I used to work in the Teton park back in 96, and I've gone to JH almost every summer/fall since then to visit for a few days. I think the fall is my absolute favorite. Less crowds, weather is cool, cheaper, leaves are turning, etc. I actually have a trip planned for Labor Day weekend, but I think I am going to cancel it. I definitely recommend staying in the park rather than Jackson. Try checking out Colter Bay Village.
As far as snow, that is so unpredictable in Wyoming. A few years back I was visiting with my mom, and it snowed in JUNE! I say go in October and enjoy yourself!!!
Torey