I agree with basically everything that's been said here, especially Joseph Weissman's remarks. I'll chime with one perspective with citations from recent philosophers who claim there is a difference between ethics and morality, and then try to point out how this difference (assuming one accepts it) would in turn generate a difference in ethics and moral philosophy (I thank Geoffrey Thomas for pointing out the need to clarify the latter--I've edited this post to accommodate the question he raised in a comment to the first version of this post).
But first, the terms "morality" and "ethics" are indeed often used interchangeably, especially in things like course titles in philosophy departments or in anthologies of essays and whatnot.
But some philosophers have recently done work to disambiguate the terms "ethics" and "morality," though the distinction argued for is often contentious--a point I'll return to below. [By "recently," I mean, "within the last half-century or so"].
Bernard Williams is an example of someone who has argued for a distinction, and it's similar to the one Joseph Weissman mentions. See Bernard Williams Ethics and the Limits of Philosophy. The basic idea here (and please someone correct me if I'm off on the details, it's been a while since I've read the book) is that ethics concerns itself with how one ought to live, and so questions like the following fall within its domain: "What does it mean to live well?", "What does a good life look like?" and even more fundamentally, "Is 'living well' something that can be conveyed or taught?" Morality, on the other hand, is more focused on a specific sort of value that our actions have, and involves standards we have to or ought to live up to along with constraints placed upon us. It tends to involve obligations we have to others, and these obligations are seen to have a special status that overrides non-moral ethical commitments, or non-moral courses of action that often imbue our life with meaning or involve things we care about. John Stewart-Gorden I think does a nice job capturing the distinction between Ancient conceptions of ethics as opposed to modern conceptions of morality: "Ancient ethics is about living a good and virtuous life according to the ethical virtues, that is, to become a virtuous person, while the modern notion of morality is primarily focused on the interests of other people and the idea of deontological constraints. That is, one acts morally because one has to meet certain standards and not because it supports one’s own good life" (https://www.iep.utm.edu/anci-mod/#SH1a). In Williams's sense, morality is a subdomain of ethics (or perhaps a distinct domain that overlaps with ethics); his argument, however, turns on the idea that morality has come to dominate our understanding of ethics in such a way that we've lost something valuable in the broader Ancient Greek conception of ethics. In order to retrieve something from what Williams believes has been lost, he argues for the distinction between "ethics" and "morality," arguing for what is wrong in particular with modern conceptions of morality that are restrictive to the point that they squelch or hinder our attempts to live well.
For the sake of clarity, his argument is not for some form of egocentric set of values--fulfilling our obligations to others may be a necessary part of living well. The issue is that the modern conception of morality is (for lack of a better term) overbearing, given that the obligations it issues are perceived as always overriding non-moral projects and pursuits that imbue our lives with meaning or contribute to a life well lived (modern morality's obligations are things we must do, and failure to deliver on those obligations makes us blameworthy). There are other reasons Williams gives, and other aspects of the distinction he draws between morality and ethics (e.g., modern morality strives for a problematic kind of systematization that is not found in Ancient Greek virtue ethics).
By "modern conception of morality," Williams seems to be referring to the rise of moral theories in the 17th and 18th centuries, such as Utilitarianism and Kantianism.
Other philosophers have argued for a similar distinction. Susan Wolf (who cites Williams) argues for a distinction between morality and well-roundedness or meaningfulness. See, for instance, Susan Wolf's "Meaning and Morality" in Proceedings of the Aristotelian Society, vol. 97, 1997, pages 299–315. Her distinction ends up seeming similar to Williams's, in that she too sees morality as dominating our ways of thinking about "the good life" or ways of living well. In another article (perhaps her most famous), "Moral Saints" (in The Journal of Philosophy, Vol. 79, No. 8 [August 1982], pages 419-439), Wolf argues for this point by using as an example a figure she dubs "the moral saint"--"a person whose every action is as morally good as possible, a person, that is, who is as morally worthy as can be" (419). She claims that such a person, though they do much good in the world, seems to be lacking a well-rounded life, and is not typically someone we would desire to emulate. As Wolf states, "In this paper, I wish to examine the notion of a moral saint, [...] to understand what a moral saint would be like and why such a being would be unattractive" (419). Regarding the figure of the Moral Saint, Wolf states: "The moral virtues, given that they are, by hypothesis, all present in the same individual, and to an extreme degree, are apt to crowd out the non-moral virtues, as well as many of the interests and personal characteristics that we generally think contribute to a healthy, well-rounded, richly developed character" (421).
So there are philosophers who think it is important to draw a distinction between morality on the one hand (typically concerned with obligations or things we have or ought to do, and is generally focused on other people), and ethics (or some broader notion a how to live well) on the other.
What would this mean for our understanding of ethics and moral philosophy?
If one believes that there is a meaningful distinction between ethics and morality, then ethics in a philosophical context is inquiry into things like: "what does it mean to live well?", "what is a good life?", and "what are the relevant sorts of considerations regarding what makes a life good?" Aristotle's ethics would be exemplify ethical inquiry as something distinct from moral philosophy. His Nicomachean Ethics is primarily concerned with: (1) "What does it mean to live well?", the (simplified) answer to which is "a life characterized by eudaimonia [a notoriously difficult word to translate, but which is often translated as, "fulfillment," "flourishing," or "happiness"--but a notion of happiness that is hard to capture (it is not happiness as contrasted to sadness or other emotional states, which is why it is perhaps best to understand it in terms of "flourishing"]; and (2) "How can one live well?", the simplified answer to which is (A) a life characterized by virtuous activity (which for Aristotle means actualizing certain dispositions or character traits that exemplify human excellence) along with (B) possessing certain "goods" (e.g., a certain degree of material wealth, such that you can devote your time to actualizing your virtuous dispositions rather than trying to find where your next meal will come from). The Stoics, to provide another example, more or less agree with Aristotle on the point about actualizing virtuous (i.e., "excellent") character traits, but differ with him on the role of possessing goods--all that it takes to live well by the Stoics lights is virtuous action (Stoics even tend to believe that should the virtuous action fail to achieve its aim, it does not detract from the value of the action, nor does that failure negatively impact the goodness of one's life--the good life just is the life of virtuous activity). Ethics--at least ancient ethics--thus often asks questions about what sorts of character traits are virtues (what sorts of character traits count as dispositions that exemplify human excellence), but ethical inquiry does so with an eye to issues of living well (moral philosophy might ask "what count as virtues?" but this has less to do with living well and more to do with "acting morally" and acting in a way that meets the demands of specifically moral obligations).
Moral philosophy would ask (overlapping) questions like, "What makes an action possess moral worth?", "What is moral worth?", "What makes an action right or wrong?", "On what basis do the obligations that a moral system issues possess their authoritative, overriding status?" and "What sorts of actions or things are moral or immoral, and what makes them so?" Theories offered by moral philosophers provide different answers to these questions. I'll provide two examples. For Kant, it is a good will that makes an action possess moral worth; what makes the obligations of morality special and overriding is that they meet the demands of certain duties we have in virtue of being rational, autonomous agents. For a Utilitarian, moral worth is determined by maximizing positive states of affairs (or minimizing negative states of affairs), or by one's best effort to maximize positive states of affairs (or minimize negative states of affairs); what makes an obligation overriding is that it maximizes positive states of affairs (or minimizes negative states of affairs) regardless of whether this incurs some degree of harm to the person performing this action (or makes this person's situation worse than it otherwise would have been). What makes a state of affairs positive or negative for a Utilitarian is hard to define in detail (different Utilitarians will disagree on these details), but generally Utilitarians operate with some conception of pleasure or happiness--so what it is you (morally) ought to do is decided in terms of increasing overall pleasure (even if that means sacrificing some of your own pleasure--e.g., donating X amount of money to help feed 10 starving families, instead of keep X amount money that you could otherwise use to buy yourself some luxury item purely for your own pleasure). Moral philosophy can also ask broader questions like, "What is a value?" or "What is it that we mean when we call something (morally) good?", though these questions begin to move into the territory of the metaphysics of value, or what's sometimes called "value theory" or "axiology." But note that moral philosophy, should one accept a distinction between ethics and morality, is not asking questions about how to live well, or what a well lived life looks like; it's bringing philosophical scrutiny to questions concerning what it is right to do.
The above distinction between ethics and morality (and the potential difference it implies for ethics as a domain of philosophical inquiry and moral philosophy) is contentious, as I noted. Some moral philosophers argue--as an undergraduate, I had a professor that made this argument--that morality simply concerns itself with what you ought to do, end of story. For him, that meant that the moral saint, for instance, is perhaps not actually doing what she ought to do if she is sacrificing well-roundedness in her own life in order to maximize doing good for others. So for this professor, morality is the broader category, and inquiry into "how to live well" is a subdomain of moral philosophy. And he thought there was no reason to suppose that "what we ought to do" always involves obligations that override all our other, non-moral concerns.
It might seem that this is just an argument about how to define terms, and in a sense it is. But for someone like Williams, and I take it Wolf as well, it tracks certain historical trends. The exemplary ethicist is someone like Aristotle, and the paradigm for ethics is Ancient Greek virtue ethics. Something seems to have changed in modern conceptions of value that gave rise to what Williams sees as morality. As Williams sees it, it's not just that ethical values gave way to moral values, it's that philosophy itself became less concerned with ethics and more concerned morality--i.e., ethics as a mode of philosophical inquiry waned as moral philosophy waxed. And Williams, in turn, uses this distinction to understand certain figures in the history of philosophy, such as Nietzsche (19th century) who pit himself against morality. So it's not just an arbitrary decision on Williams's part to draw this distinction--he thinks it tracks something in the history of philosophy, and uses it to interpret certain trends, movements, changes, and figures in the history of philosophy.
As a relevant side note, it is indeed true, as Bob notes, that "ethics" (from Ancient Greek) and "morality" (from Latin) are both based on terms that in their original languages mean something like customs, habits, conventions, or the like. But it is worth noting that Cicero (a Roman), when translating Greek works, decided to translate the Greek "ethikos" into Latin as "moralis" (see https://www.etymonline.com/word/moral and https://www.iep.utm.edu/anci-mod/#SH1a).
While I don't remember Williams arguing for this, I do recall reading (though I can't cite a source, my apologies) that this point has historical importance: Despite the similarities in meaning between the Ancient Greek "ethikos" and the Latin "moralis," Cicero's works were the go-to in the Roman world for all things Greek (at least all things Greek philosophy). So Cicero makes this translation decision, and many centuries later, as Roman religious practices wane and Christianity comes to the fore, Christianity picks up the term morality (or rather, "moralis"), and over time, morality turns into the sort of system of values Williams sees it as--leaving "ethics" to refer more exclusively to Ancient Greek conceptions of value in terms of a life well lived (connected to certain complicated notions of happiness or "eudaemonia")--until for whatever reason, ethics was picked up again more recently as either a synonym for the (at this point changed) meaning of morality, or as applied moral theory (as in "business ethics" and "biomedical ethics").
For someone Williams, ethics as a mode of philosophical inquiry into varieties of living well has been overshadowed by the concerns of moral philosophy, and the original meaning of ethics, and the inquiry that defined it, was lost; moral philosophy and its concerns assimilated the term ethics without assimilating the mode or object of inquiry that originally defined it. This is why moral philosophy and ethics are often taken to be the same pursuit, and why someone like Williams was motivated to disambiguate ethics from morality (since to do so would be to retrieve ethics as a distinct mode of philosophical inquiry with an object different from that moral philosophy).
And last side note, with respect to Frank Hubeny's question (March 30th), Aristotle claims that ethics is the domain centered on how individuals can live well (and what it means for them to live well), and politics is about this question applied to collectives (to the polis, or city-state); he actually sees politics as the larger, more important category, and ethics as (in some sense) a subdomain of politics, because the focus of ethics is smaller.
[Thanks again to Geoffrey Thomas for constructive criticism, which I hope this edited response has adequately addressed.]