Within contemporary philosophy, "logic" is narrower than "reason." We can waste our time in etymology, but the key is how the terms are used in the discipline.
In most* contemporary philosophy, "logic" refers to sentential logic, truth-predicate logic, modal logic, deontic logic, and other variations on the formalization and evaluation of arguments in this way. The adjectival use "logical" has a broader meaning that encompasses things authors think are close that level of rigor. (See for instance The Journal of Philosophical Logic).
I qualified the above paragraph with a most. Allow me to explain. There's a supposed divide between "continental" and "analytic" philosophy (here I will sketch it briefly just to relate it to the point). It's much touted by some who write about philosophy (rather than philosopher per se), and it's beloved by two groups within philosophy. First, it's beloved by a group of self-styled "continental" philosophers who use "analytic" pejoratively to refer to people doing the sort of metaphysics that grew out of logical positivism, philosophy of language, formal logic, etc. Second, there's a mirror group of "analytic" philosophers who use the term "continental" philosophy to refer to word salads and grandiose claims made by people who do philosophy with less "rigor." For the most part, this distinction does not make too much sense. We can draw some fuzzier lines by pointing out that most Frege, Carnap, and Russell scholars are going to be in the anti-"continental" camp and that many Derrida, Hegel, Levinas, Foucault scholars are going to be in the anti-"analytic" camp. But this is far from an absolute test... This long digression to say that among the "continental" hard liners (as Brian Leiter calls them), here are some people who will use "logic" and "reason" in ways that are unlike the breakdown suggested in the answer. (As a counterexample, Kenneth Westphal uses the terms in the "standard way" and is a Hegel scholar; Brad Stone also publishes using the standard distinction and does Foucault and Heidegger).
"Reason" in contrast is going to be harder to define for contemporary philosophy. Reason is the name of a faculty in modern German philosophy that is above and beyond understanding. Thus, Kant's Critique of Pure Reason is a critique that uses reason to critique the claims of the understanding and seeks to delimit the function of understanding (and to a lesser extent the function of reason). For Hegel, Reason is the more important faculty.
In a separate stream, reason is used in a manner pretty close to any dictionary usage to refer to our capacity to think. This, often, is too imprecise so the generic capacity is sometimes called "ratiocination" (in part because that word is so obscure that no one brings pre-conceptions to the table with).
Reason is also an important concept in medieval philosophy from which the modern German philosophy inherits and develops the concept. For the medievals, logic is a propaedeutic to the study of philosophy (i.e. reasoning) which precedes the ability to do good theology.
In ancient philosophy, things are really confusing on this front because as ig0774 points out in a comment the Latin word ratio is the most common translation of logos with philosophy. So thinking about the neo-Platonists, there's not really a difference except when one is manufactured to make a point there...