This is my opinion, based on personal experience:
Even though there are some good deals right in the middle of Summer, it's much too hot! I don't want to be hot and miserable while on vacation if I don't have to be and planning to go smack dab in the middle of Summer pretty much guarantees hot and miserable. June and July are the busy times so hot + humid + crowded = grumpy me!
I don't like traveling in Winter. We are in Pennsylvania and when we go to WDW, we drive. I don't want to chance the possibility of a snow storm on the way there or back. Winter also seems to be the time of year we are all sick - off and on. I tend to stay away from vacationing between the months of November - March, simply because of this.
Of course, staying away from WDW during Spring break and holidays makes the most sense, since that is when many people go - because they can. My oldest is in Kindergarten and I quickly found out how strict the school district is. The students may not miss the first 2 weeks of school, so going right after Labor day is out and the students may not miss finals, so the beginning of June is out.
I have been to WDW in February - I was in 3rd grade so I don't remember much about the crowds but I do remember that the weather was a bit cool and we didn't go swimming in any outdoor pool, even though they were heated. I have been there in early June - It was a school trip! It was super crowded - YUCK!
I went in late September and it was AWESOME! Weather was beautiful - we didn't experience any storms (hurricanes) and the crowds were practically non-existent. I think the most we waited to get on a ride was 20 minutes!
I was there in May, around Mothers' Day and it was WONDERFUL! So far, in my opinion, that was the best time to go!
From the end of April to the middle of May is the perfect time. Right after Easter and before Memorial day, it's like you're getting there in between the crowded times. Now, yes, this is my opinion and I am sure others will have their own 'favorite' time but for me, I don't know if I'd bother going any other time. Last year we went from May 11th - May 17th and the weather was in the mid 80's during the day and high 60's at night. PERFECT! We took advantage of the pools and water rides! The crowds were not too bad either. We never had to use a FastPass. The longest wait, that I remember, was about 35 minutes for Expedition Everest and that wait was worth it!
This year when we were there from April 29th - May 6th it was once again, awesome!