Well, if you go in August after the Florida schools have gone back, then it is less crowded than the end of June, but hotter, at least when we have gone.
I'd say end of August is better. While it might be technically hotter than late June, you're talking about a degree or so and either way it's plenty hot!
But the crowds in late August have to be substantially less than late June, I'd think. So I'd go at that time if a choice between the two.
My vote is for August. Florida schools usually start the third week of August. We were there this past August for ten days with smaller crowds. Lots of Brits on Holiday. Whew! The weather though! Stay hydrated whenever you go!
We've gone several times in June, July and August (all different weeks). If your goal is less crowds, we saw a considerable difference in August. But, man...it's hotter than a cat on a hot tin roof!!!
DH and I made the comment that we didn't think the heat was THAT much worse and for the differences in the crowds, we'd try and plan August for future trips if we go to Dis in the summer.