Seasons are there to make the game look more realistic. There are some different things depending on the season you're in.
And much more, check the link below for more info.
- You can reach different places depending on the season. Fore example, you can get to the ledge in Winter in Irricus City that you can't get to in the other seasons
Different seasons mean different Pokemon to catch. (Like, Spheal only appears in Winter)
You can battle Cynthia during Spring and Summer.
More info on Seasons