It will make zero difference for any cable manufactured in the last 15 years, guaranteed.
(I am taking the p***! A network cable is a network cable)
Yes, there are differences. You just WILL NOT SEE them in real world conditions with a games console. Switches, hubs, repeaters - hard and fast rule, they either support 100baseT or 1000baseT, yes. Cables? Not so much. With a short cable run between two gigabit devices cat 5 vs cat 5e is very very very unlikely to make any difference at all. I think it very unlikely anyone still has Cat3 cable around