Early June is VERY HOT and very busy. Rainy season usually starts in May so you'll get that rain as well. In comparison to early August, I believe the crowds will be the same if not more crowded.
We actually prefer mid-late August over early June because the sun is not as intense as it is June. It just tends to be more overcast in August than June and that helps filter some of the sun's rays,. but the humidity level is about the same. Of course, in late August there is definitely a decrease in crowds, but increase of hurricanes and tropical activity. Early August is definitely still busy.