Humidity wouldn't be as much of an issue as July and August as already mentioned but we've experienced serious rain on trips during June which has often taken the shine off our trip. In 2005 we had 8 days and nights in Orlando before heading to the Keys and Fort Lauderdale and we had rain pretty much for most of the day for at least 6 of those days in Orlando. You will often hear people commenting on how the rain in Florida comes along mid afternoon, chucks it down for two hours and then clears up leaving a pleasant dry evening, believe me this is not our experience but of course other years may well have been fine, but certainly in 1999 (our honeymoon), 2003 and 2005 which are the three times we have gone in June it has rained considerably during the early-mid part of June.
When the sun does shine it is all rather nice although very hot. The parks are quieter although you will still find crowds on major attractions especially water rides if the sun is beating down. In 2005 Splash Mountain was 100mins wait. By contrast, when we went in October/November a few years back Splash was walk on.
Personally, we've agreed that June is out from now on in Florida as the weather can ruin a two week holiday and would consider May (although it rained a lot during May in 2009 according to people I know who went) or later in the year with late October, early November being particularly pleasant with temps around 26-28c, little to no rain (only had one rain day out of 11 in 2006) and the parks virtually empty.