Hi Carrie,
Welcome to the Disney Parks Moms Panel! I was excited to answer your question when I saw the dates you were considering, because I visit with my own family around the same time every year. Our preference is to arrive the weekend after Memorial Day is observed to avoid some of those holiday weekend crowds, so for 2014 that would be the week of June 1, 2014. It's been my experience that summer crowds are just starting to arrive at Walt Disney World, but aren't yet in full force since many schools around the country haven't yet let out for the summer. This is just my opinion, of course, and I do think you'd have similar crowd levels either week.
As for the weather, the average high that time of year is around 90 degrees and the average low is around 70 degrees. June is usually the rainiest month of the year, so I'd plan on packing some ponchos just in case you need them.
Also, I can't let you go without mentioning one of my favorite Walt Disney World events of the year. If you're a fan of Star Wars, you'll be able to enjoy
Star Wars Weekends at Disney's Hollywood Studios no matter which week you pick.
Happy vacation planning, Carrie, and please let me know if you have any other questions!
Beth