We have been to disney in June several times. In general:
The first week of June is not terribly crowded for summertime standards as most of the country is still is school.
The second week will start to become more crowded as more schools are letting out, but still very manageable.
By the third week in June, you will notice it is becoming quite crowded at the world as most people vacation at this time of the year. The peak summer season has begun at disney and will remain through the end of July at this point.
It will be hot, hot, hot at disney in June, period!!! There is no way around it....expect temps in the mid 90's for the most part, humid, and it WILL rain most days! most likely in the afternoons; but the occassional all dayer. But, it will only be worst in July. So, keep this in mind if these are your only two time periods to choose from.
You have to decide if your family is suited to handle these types of conditions while on vacation.
After saying this; don't be too discouraged. With the use of EMH, fast-passes and taking breaks from the midday heat; a june trip to disney is not only manageable, but very magical.
We were there over easter week this past year and found it very manageable!!! Although we have decided not to go that week ever again(too short, not too crowded)! remember: hot and crowded at disneyworld is still better than just about anything else!!!! go and enjoy.