We have gone during both seasons, and here's my experience and advice:
March: Comfortable, but very unpredictable weather--you may have some summer-like weather, or some chilly weather. Likely, you'll have something in between. Generally dry. This is usually a great time to tour the parks without getting really hot, but not necessarily a great time to go to the water parks or swim/lounge at the pool in your bathing suit if the weather happens to be on the chillier side when you're there. Flower & Garden festival going on at Epcot. Mid-March is also one of the highest priced times to go to Disney as many people are on spring break, or looking to escape the winter and soak up some Florida sun.
Early June: Hot, humid weather, but not necessarily as hot as it would be once you get deeper into the month. It will likely thunderstorm each day mid-afternoon. I'd expect temps ranging anywhere from upper 80s to low 90s. Great swimming weather; not as comfortable touring weather (depending on how well your family tolerates heat). Star Wars Weekend going on at Hollywood Studios. Parks a little less crowded than they would be starting the 2nd week of June and going deeper into the summer. Prices are lower than March.
For our family, we feel a little incomplete without getting a chance to really soak up some sun and get some decent pool time in at the resort, but that is also because we go to Disney several times a year, and riding all of the rides is not our priority. My kids also like the waterparks just as much as the theme parks. So if our family had only one time to go a year, I'd pick summer. BUT, if your priority is going commando throughout the park and riding everything, and pool time is not as important, I'd pick March. It's much more comfortable, and likely (hopefully) you'll have at least one kinda-summery day to spend a little time at the pool.