Late September was really, really nice for us. It was hot but not OH-MY-GOSH I'm dying hot, just kind of let's take a break and swim hot. And the crowds are non-existent.
You do get the daily rain storm, but hey, that's Florida.
We just went the second half of Presidents Week. Now, I can't recommend this, especially to first timers, because the weather can be hit or miss and crowds are unpredictable, but-- we lucked out with GLORIOUS weather, 70's every day, sometimes in the 80's, and only one rainstorm. We even found the crowds managable. Don't get me wrong, there WERE crowds--- but we had a plan and they didn't bother us much.
I find that I really need a vacation in the dead of winter and Florida really hit the spot.
Next vacation for us will either be late September again, early May, or November.