My inclination is to say that it doesn't, and that the disagreement is really just about how to use the word "exist". But there are a couple of ways in which the distinction might have a bearing on anticipated experience.
One prominent argument for Platonism is the Quine-Putnam Indispensability Argument, which says that our best strategy for ontological commitment is to believe in the existence of those objects over which our best scientific theories indispensably quantify. So if one cannot dispense with quantification over mathematical objects while maintaining the integrity of our best theories, then we should believe in the existence of those mathematical objects. If one accepts this criterion, then whether or not one believes in the existence of abstract objects depends on whether our best theories indispensably quantify over such objects, so the Platonism vs. nominalism debate depends on science. Hartry Field wrote a book where he tries to axiomatize Newtonian continuum mechanics without any quantification over real numbers.
Also, it's plausible that Tegmark's Level IV multiverse hypothesis assumes Platonism, since it requires that mathematical structures have independent existence. So if you believe Tegmark's hypothesis constrains anticipation, then perhaps Platonism does as well.
Hartry Field wrote a book where he tries to axiomatize Newtonian continuum mechanics without any quantification over real numbers.
Ah, that sounds a bit like coordinate-free physics.
Despite being (IMO) a philosophy blog, many Less Wrongers tend to disparage mainstream philosophy and emphasize the divergence between our beliefs and theirs. But, how different are we really? My intention with this post is to quantify this difference.
The questions I will post as comments to this article are from the 2009 PhilPapers Survey. If you answer "other" on any of the questions, then please reply to that comment in order to elaborate your answer. Later, I'll post another article comparing the answers I obtain from Less Wrongers with those given by the professional philosophers. This should give us some indication about the differences in belief between Less Wrong and mainstream philosophy.
Glossary
analytic-synthetic distinction, A-theory and B-theory, atheism, compatibilism, consequentialism, contextualism, correspondence theory of truth, deontology, egalitarianism, empiricism, Humeanism, libertarianism, mental content externalism, moral realism, moral motivation internalism and externalism, naturalism, nominalism, Newcomb's problem, physicalism, Platonism, rationalism, relativism, scientific realism, trolley problem, theism, virtue ethics
Note
Thanks pragmatist, for attaching short (mostly accurate) descriptions of the philosophical positions under the poll comments.
Post Script
The polls stopped rendering correctly after the migration to LW 2.0, but the raw data can be found in this repo.