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.