The games were dedicated to Zeus. This went before the squabbles of mortals. Whatever their enmities, all Greeks agreed that Zeus ought to be honoured in the proper way, and that there was great glory to be had for the winning athletes and their sponsors.

That said, they didn't quite celebrate the games "like nothing happened". In the Archaic period, Elis and Sparta fought wars over who had sway over the town of Olympia and consequently got to host the games. In later times, Elis was not above excluding Sparta from participating in the games unless some running grievance was redressed. Also, since the Games brought together representatives from all corners of the Greek world, they were also fantastic opportunities for back-room deals and impromptu conferences between states.