Thought I would weigh in. Sept and early Oct, can be snow or the most pleasant weather you can experience. At this time the elk are in the rut and the meadows turn to gold. Earlier in season, May through June you will see many of the newborn animals. They are cute and they play. Great time to be here.
I am in Yellowstone now. Winter is magical. Not for an RV, although there are RV parks open in West Yellowstone, so some do it. No crowds but expensive to get in on a snow coach.
Attached is a picture of the Visitor Center, where the Dicks and Jolene and I work, along with one of the over snow vehicles, (operated by AlpenGuides), which are the main way of travel in the park.
Personally I always go in either July, July or July. Thought you would have learned. learned. learned...