There is an important and very practical difference when it comes to ethics (as in "how to live your life properly") and the consequences of either way of thinking when used as axioms.
Platonism holds that ideas do exist by themselves and are the purest form of existence. Things and living beings are but mere approximations, imperfect realizations of those ideas. For exemple, the idea of a Person exists, and as the purest form of person there can be as all the virtues that we might expect of a person. We on the other hand are only imperfect embodiments of this idea, just like a square we draw on paper is an imperfect representation of the idea of Square. The sides are not exactly straight or equal, the angles never perfectly 90 degrees, in the same way that we will never be as honest, as courageous or as wise as the perfect idea of a Person.
As a result, under a Platonist view, we humans are always lacking, always imperfect. All our self improvement can bring us closer to the idea of a person, but we will never reach it. Now add to this that the ideas of gendered Man, Woman, Citizen, Soldier, Worker etc do also exist and you can see in how many many dimensions we are lacking, never good enough (it can seem terribly self centered to demand that people conform to OUR idea. But keep in mind that a perfect idea under idealism is not a mere opinion, but universal truth that was reached by the careful application of reasoning).
On the other hand nominalism posits that our idea of a person was constructed out of the many individuals we met that could be clustered into the category of "person". Individuals exist before the idea, they are what really is.
As a result, when an individual does not match our idea of a person, it is not the individual who is lacking, but our idea, our preconceived notion of person, that is inadequate. As this idea exists only in our thoughts, individuals that don't match with it can't be held responsible for this discrepancy. Obviously, it is our idea that has to be changed in a way to match new data.
(About the concept of inadequate ideas, see Spinoza's Ethics. An inadequate idea is roughly speaking an idea that does not match reality.)
So why care? We can see that, depending on our ontology, our expectations toward others and ourselves change dramatically. Let's say you promised yourself to stop smoking but can't help having a cigarette from time to time. Are you an imperfect embodiment of the ideal Person, lacking in willpower and wisdom, and should you kick yourself mentally until you reach this standard of perfection? Or is it just a fact of life, that you overestimated your commitment and should reconsider if you really want to stop smoking, or change your strategy by avoiding temptation in the vicinity of tobacco or smokers?
Idealism can be motivational by setting for ourselves high goals, but also a lot of stress as we constantly fail to match expectations. It also shapes our view of others, setting high standards for the people around us but also leading to a very demanding attitude, as we expect others to comply to our vision of what they should be. Roughly speaking, it accords itself better with deontology or virtue ethics: since there is but one true idea of what a proper person is, it's only logical that everybody should strive toward fully realizing this idea. We and other people are responsible for not conforming to the pure, existing idea of what we should be.
Nominalism on the other hand leads to an approach of ethics much less demanding but also more free. It can lead to whataboutism and a slacking attitude, after all we are what we are, why try and improve toward a goal that has no real existence? But also it leaves us free to fix our own goal, our own vision of what we ought to be (See Sartres' existentialism. Nietzsche also comes to mind). It also leaves each of us responsible for revising our standards in accordance to reality: if people don't act as we expect them to, it's not them that are imperfect but our expectations that need revising. A nominalist ethic would probably tend towards consequentialism, as it is more difficult to argue for the existence of a universal idea of duty if ideas dont exist by themselves.
For a real life, current example of why it matters we can look at the public debate about the notion of gender.
The idealist approach is to posit that there are the idea of Man and Woman, that those ideas are eternal and fixed and that people who do not conform to those standards are lacking in some way: delicate or passive men are imperfect, effeminate Men, butch or bossy women are imperfect, boyish women.
A nominalist approach posits that the ideas of man and woman are constructed and that people who do not conform to it are not at fault, that it is those standards that should be reconsidered or entirely abandoned. Some men just happen to be delicate and some women butch, and we should not expect them to be other than they are.
If you pay attention to the talking points on both side of the controversy, you will notice how each is heavily based on one of those opposed ontological axioms (even if they don't make it explicit or realise it by themselves) and would be put in extreme rhetorical difficulty if their own axiom was to be denied.
Another example is politics and the Law. Does the set of perfect laws for the city exist by itself somewhere, and then what we call politics is just a way to approximate this perfect legislation? Or are laws just a social construct and politics a way to reach a consensus toward a reasonably efficient set of rules? The former would tend to favor aristocracy (in its original, platonic, meaning of "rule by the best, the wisest"), while the latter would tend to favor democracy, as the more citizens participate in lawmaking the more it is possible to reach a consensual result.
And that is why we should care about the nominalism VS idealism debate.