Early June will have less people, but chancier weather and hiking conditions.
August would be nice weather, but lots more people. If you like hiking, even short hikes, I'd spend at least some time in Grand Teton, not just drive through.
I would say that in August reservations are a must, and do it now. I don't know if early June fills up that much, but reservations now is still not a bad idea even for early June.