Perceived Pros:
1. A much more stable riding platform compared to two skis.
2. Better for jumping ( I love hitting those small ramps on the side of corridors and trails

)
3. Larger surface area - can handle deep, fresh powder better
Perceived Cons:
1. Can't pole on flat terrain like skiers, must walk or push with back foot.
2. Strapped in for better or worse. Hello wipeout, goodbye ankles.
3. Harder to control in tight quarters.
Edit:
I've done both and prefer old fashion skiing. It's hard to find sensible people on the mountain anymore. Etiquette is rarely taught in ski/boarding lessons. Laying down below the crest of a drop off... that's one way to get seriously hurt. If you're going to lay down or stop on a run, move off to the side and stay visible.
BTW,
Claimjumper and Duke's Run, my favorite runs at Breckenridge.
Oh yeah, I know I would never do that, but I always fear that I'll be the one that comes flying over that crest and smoke a downed skier or little kid. Claimjumper sounds familiar, don't remember Duke's run but I'll be sure to check them out. We'll be at the Beaver Run Resort.