Keep in mind too that if you're looking to watch the animals, May is a terrible month for that. Most are still coming out of their winter phase, coats look bad and a lot of them are skinny due to the stresses of winter. Compared to what they look like later, mid-June through the end of the season, even with crowds, the animals are much better to look at by then.
If your focus is just on scenery, then it doesn't matter really. Go earlier in the season to avoid the crowds but, some trails and roads may still be closed. Remember, Yellowstone is pretty high in elevation so winter leaves much later than most other places.
I personally enjoy the fishing most. July is really the start of the best dry fly fishing in the park. Salmon flies hatch in the the streams in the park later in June, depending on various factors. Lower-down, toward Bozeman, they will sometimes hatch as early as Memorial Day but usually, early June is good timing for that hatch around Bozeman. Working their way upstream to the park from there.