I lived in Steamboat for two years and then moved to Ogden, so I've ridden a lot in both areas. The other posters have it broken down pretty well for you: Steamboat is great for trees, but really lacks any good steep, open terrain (or open terrain period). Since most of Steamboat's terrain is cut runs, it tends to get tracked out more quickly on a powder day. Of course, that's why you spend the rest of the day(s) ducking into the trees.
Snowbasin is quite the opposite. It's got a ton of open terrain and you can find untracked lines for days after a storm without really looking too hard--not necessarily top-to-bottom runs of untracked, but solid swaths. The tree skiing isn't nearly as good as Steamboat (though there are plenty of nice trees of off John Paul and the Tram), but there's infinitely more steep and bowl terrain. Snowbasin also has a ton of sick backcountry and out-of-bounds terrain that requires little or no hiking. Steamboat, not as much.
Powder Mountain is gigantic and is worth the trip just for the fact that it's such a unique resort. With its tow lift/hike-to, shuttle pick-up and $15 snowcat access, it also holds powder stashes for days. My general formula is to hit Snowbasin when it's snowing and Powder the day after. In a lot of ways Powder reminds me of Steamboat: not steep at all, great snow, and perfectly spaced aspens for tree skiing. The tree skiing isn't quite as deep or good as Steamboat due to the limited vertical (around 2,000 feet) but you won't really be thinking that while you're skiing it.
I'd disagree with other posters that use snow or nightlife as factors one way or the other. Utah and Steamboat both have killer dry snow. Utah gets more and Snowbasin and Powder will have untracked terrain long after the 'Boat. For some reason, I've seen this idea popping up lately that Steamboat has some major nightlife to speak of. Bars, yes. "Nightlife," not so much. Ogden probably has more bars and nightclubs than Steamboat, though they're more spread out and not quite the apres-ski atmosphere that you'd find in a smaller ski town. There are also a few strip clubs in Ogden if that's your thing (or if you want to make it your thing).
Steamboat is definitely a resort while Powder and Snowbasin are basically ski mountains with the main town about 20/25 minutes away. Neither has a base village and Snowbasin doesn't have a single bed at the mountain. That's great as far as I'm concerned--keeps the crowds and tourists elsewhere. But, on a ski trip, you may prefer more of a resort experience.
If it were me, I'd choose Utah every time, particularly if you like tasting several flavors of resorts (Steamboat is pretty isolated, unless you count Howelsen Hill). But, you really can't go wrong either way.