September is a great time to visit Yellowstone. You will probably get some snow, but there wont be any bugs. I think the reason they close the campgrounds so early is that they don't plow the roads in the park if it does snow. The roads will normally clear pretty quickly after a snow. The Elk will be in the rut and the males will be bugling. It is pretty neat hearing the serenade. You will see more wildlife because the roads are at a lower elevation than where animals spend the summer. You will see herds of deer, lots of Buffalo
The Lamar valley will be full of animals. You will find one of the more successful packs of wolves there. You won't see bear as they will all be in the wilderness areas that surround the park feeding on gut piles. (Think Elk hunting season.) In the back country don't be surprised to see park rangers packing M-14s
protecting against poachers.