Former 31 year resident of Estes Park and professional guide in Rocky. Most points have been covered. But I'll put my two cents worth in. In heavy snow years many trails are still snow packed in early June which is bad for bad knees. Trail Ridge Road can close due to storms. Crowds are not as bad. Summer flowers are not out yet. If you want to stay in the Park, reservations are needed. In dry years last two weeks of August can be pretty brown. Summer flowers mostly gone. Weekday crowds are not as bad as school has started or about to. June would be my pick.
As for Grand Teton, there are many short loop hikes along the base of the Tetons. These are flat which is good for bad knees and the views are nice. Also, a raft trip through the Park is worth a day. Do the early morning trip since you have a better chance of seeing animals. There just aren't many easy hikes. The ride up the ski gondola is worth it since you can catch a couple trails from the top.