We have been at Disney the 3rd week of June on at least 9 trips (it's the week that the students and teachers in our family are finally all out of school). In general, we're ok with it. We plan for the weather -- taking breaks at the resort in the afternoon, being prepared for rain, etc. There are some years that it has been ungodly hot -- highs in the upper 90's and high humidity. We would sweat just walking out the door of our room/building in the morning. We try to do indoor attractions throughout our time in the parks to get a break from the heat. But we've also had years where the temps stay in the mid to upper 80's most of the week and humidity is tolerable. On our trip in June 2024, we had one day where it was a little drizzly/misty in the morning (before 10:00 a.m.) and cloudy all day (but no real rain) and the high temp that day was 81 -- it actually felt a little chilly to me. Rain hasn't really been an issue for us on our June trips. Some trips we see the typical FL shower some afternoons and some trips we have a few hours of rain one day. Some trips we've had zero rain during the day, but one or two overnight showers during the week. And it can be raining at Magic Kingdom, but bright and sunny at Animal Kingdom. We've never had enough rain to feel like "it rained a lot" during our June trips (but I know we've been pretty lucky with rain during our trips). The only trip we had a lot of rain was the year we went in May. It rained for hours on end every single day -- an inch or two every day. We bought the Disney rain ponchos (the cheap dollar store variety wasn't cutting it in the monsoon like downpours) and soldiered on. Parks were empty, we got a lot accomplished, and still had a great trip. One of the advantages to going in the summer is that it has slowly become less crowded and the parks are open later each day. For us, June works and I've never regretted taking June trips.
All that being said, if I had the opportunity to go in April, I'd jump on it as long as it was at least 7-10 days after Easter. I would just like to experience the parks in Spring when it's a little cooler than June.