I'm late getting in on this, and must say that most of you are correct. There is no perfect time to visit Yellowstone because the park is dynamic and constantly changing. What you want to see and do can greatly influence your best time.
Many locals visit the park just after the roads are plowed and the gates open in late May -- in part because bears are often easiest to see at this time as they feed on winter killed game. The roads are relatively free of cars then too, but the trails are often still closed because of the snow. All campgrounds are not open then either.
Early June remains a nice time. Animals are still plentiful at lower elevations and the roads are still less busy as many families are still waiting for their kids to get out of school before a visit. At lower elevations, the wildflowers are just beginning to appear.
Mid to late June and early July is when the wildflowers are at their peak display. Birds are still in nice breeding plumage and plentiful. Elk generally will have started off to higher elevations and thus are harder to find. Bears are at the same visibility level they are most of the year -- which means you may not be in the right place at the right time and never see a bear, or you might just happen to have a ringside seat to a sow and cub digging away on a hillside. The small mammals such as the ground squirrels are very visible and often overlooked by visitors searching the distant slopes for the small dot of a bear or wolf. In early July, wolf pups can make their appearance outside of a den and some years make for a great show (and huge traffic/parking dilemma.) Waterfalls and cascades are usually going full blast at this time and well worth the trek off road for a visit.
Late July and August will find the largest crowds on the roads and in the campgrounds -- but will also have the warmest weather. All streams will be open now for fishing. Later season wildflowers will be blooming, but the show is less spectacular than earlier. Bison are still abundant and easy to see in the Lamar and Hayden Valleys. Many of the thermal features are easiest to see at this time because their steam is less present than in cooler months.
is one of my favorite seasons in the park. Roads are less crowded (though the fall
numbers of people are now higher than they were in the past). In early October, the Elk have moved to lower elevations and the males have begun their bugling. Several large males and their harems are usually right in town at Mammoth Hot Springs and you'll rarely have a chance to see these large animals this close other than in a zoo. (But every trip into the park will show you some stupid tourist getting far too close to an elk, bison or bear for the extra special photo, so follow the rules and keep your distance.) Nights are cooler, commonly dropping below freezing and days too can be cool. Aspen are often at their peak color around the mid to lower elevations of the park near the first of October.
Thermal features are generally nice all year. They don't migrate to higher elevations nor do they hide from the tourists. As someone else has pointed out, the mud features are usually less dynamic than early in the season, but there is still a ton of hot water and steam to see everywhere.
I disagree about the warning to not camp in a tent. The park is filled with people camping in tents in the front country campgrounds and backcountry sites all summer long. Yes, there are a few sites in and near the park that are "hard sided only", but most campgrounds will have tents and tent trailers in use. You are more likely to get killed or hurt driving on the roadway than a bear bothering you in a tent. Falling into the Grand Canyon of the Yellowstone or another of the steep canyons has killed too many people in recent years, so watch your footing here. Nevertheless, you should choose the type of camping you are most comfortable with. I suspect most readers on this site will have a panel of fiberglass between them and the outside when they sleep.
Yellowstone is really a great place. It does not have the sweeping vistas of jagged peaks like the Tetons to the south, but it has beautiful rivers, fantastic open valleys, great thermal features, many animals and great flowers. There really is no bad time of year to visit -- just different.
Sorry I rattled on so long.