I went July14-24 2005 and Jun 8-12, 2006. June is definitely better. It was extremely hot last year in July. The first couple of days started out ok, then it got to hi 90's lo 100's for the rest of the trip.
June of this year was pleasant the entire time. They have June gloom where the mornngs are cloudy and overcast, but it burns off by late morning. The week I went in June switched to later park hours to than the week before???? Note that Thursdays are Grad Nites and DL and DCA will close earlier.