Our family has made 6 trips to Disney World in mid-late June over the last 15 years. We always manage to have a great time. It's been 30+ years since I've been there in March or April so I can't really compare to those trips. And we were at Disney in May 2009 hoping to experience a little less heat and lower crowds but it poured down buckets of rain the entire week, so I can't really compare June to that trip either. So, all we really know is Disney in June. Here are my thoughts based on our experience.
1. It's often hot and humid, but temperatures can really vary. We've had weeks where the temps never got over 88 degrees for most of the trip and we've had trips where the actual temps were 100-103 every day. Many days have been in the mid 90's. We just assume it's going to be hot and humid every day we're there. If it's not, we consider it a gift. Because of this, we try to do rope drop and stay in the parks until early afternoon. Then we head back to the resort to escape the heat and crowds. We return to the parks in the evening and stay until closing time (or close to it).
2. We have a plan, but we also stay flexible. We try to knock out some headliners during the first hour or two after rope drop and schedule our FP's for approximately 10-11:00; 11:00-12:00; and 12:00-1:00. If the park isn't too crowded or it's not too hot yet, we may stay a little longer, otherwise we head back to the resort for a swim or just to hang out in the a/c.
3. We've never had a ton of rain during a June trip (however, that trip in May was a different story). There can be a late afternoon shower, but we've rarely had long periods of rain. And if we do have a 2-3 hour rain event, it's usually just once during the week. I carry a cheap rain poncho in my pocket every day because I hate walking around with wet clothing. However, my husband and kids have no problem walking around with wet clothing so none of them ever bring a rain poncho to the parks (unless there is a long rain event in the forecast). We've had trips where we've seen less than an hour of rain during the entire week. On our trip this year, we only had rain 1 day. It was a 2-3 hour event but we were in our room packing up for our early departure the next morning so it didn't really bother us. By the time we were ready to head to MK, the rain had slowed to a drizzle and the temps had dropped to the low 70's. It actually felt cool.
In general, my advice is to have a general plan for each day -- take advantage of early mornings and late nights if you can. Be prepared for the rain, but don't worry about it. Understand that it's going to be hot and humid but plan to be back at your resort or doing indoor things like shows or long attractions (think Carousel of Progress or Ellen's Universe of Energy) during the hottest part of the day.