I just returned from skiing Snowbasin and Powder Mountain. Here's the perspective of a dad with three teenagers. As background, we're all comfortable skiing up to single black diamond slopes at both resorts. We all enjoy exploring, aren't big on terrain parks, and prefer trees to trails. We despise long lines and crowded slopes. Because of this, both resorts are on our must-ski list when we visit Utah.
I mostly agree with the previous posts, so will comment on a few other aspects of the two resorts.
Some differences we noticed... fresh snow tracks out much faster at Snowbasin. Their lift capacity is light years beyond that of Powder Mountain. The good part is you can cover a lot of terrain in a day. The flip side is everyone else does too. We put down fresh tracks at Powder Mountain a day later, while Snow Basin was pretty much tracked out a few hours after the lines opened.
The food. I know a lot of folks couldn't care less about this, but for those who do, here's my take on the two. Resorts don't have to offer fine cuisine, but they shouldn't pawn off a $10 piece of gristly cereal patty as hamburger either (that would be Snowbird). Powder Mountain is a fun loving brats 'n beer crowd. It was warm, so we sat in the snow, dined on charcoal grilled brats, and listened to the tunes. It was very much a party atmosphere with the powder mountain folks going out of their way to make us feel welcome. Enough so that we came back two days later. Decent food, friendly people. The chili can run out, as we found out later that week. The burgers were very good.
Snowbasin cuisine, in my opinion, is the best. Food at any resort is expensive, but here you get what you pay for. Tomato bisque? Excellent. Pizza with gorganzola and fresh sausage? No problem. Where else can you relax and dine with dripping ski boots while enjoying the world-class architecture and incredible views (grizzly lodge). The total cost for a meal is the same as any other resort, but these guys are real chefs and take pride in their food preparation and service.
As others have said, If you're in the area for any duration, try and catch both places. If it's a powder day, ski Snowbasin first, then Powder Mountain.