Last year was my first year patrolling so take what I say with a grain of salt. Skier_j seems to have some good thoughts in his post above.
I used AT boots for a number of reasons and found some good benefits I wasn't expecting. First of all, its great to switch them to walk mode when working a wreck, but you better be damned sure you put them back in ski mode before you jump in the horns or on the tail rope of the sled. Its disconcerting trying to ski and edge hard with a heavy sled when they are in walk mode (the good news is you usually get a chance to stop and switch pretty quickly - unlike when I made this mistake at the top of a narrow chute when backcountry skiing in the spring - once I started there was no stopping until I was out of the chute - bummer). AT boots are pretty comfy and light and so all the off-ski work stuff is much easier. Its also much easier to walk in the parking lot or on rocks with the articulating cuff and good soles. I watched a couple of seasoned patrollers slip and fall between our patrol room exit (comes out to the parking lot) and the snow when carrying heavy crap. It would suck to get hurt in the parking lot.
I had both Duke AT bindings and Dynafit bindings on the skis I used to patrol. The Dynafit ones are OK, but can be a bit challenging to get in/out of when you are in strange terrain and don't have ski poles. They are very light which is nice, but its definitely easier when you can jam the tails of the skis into the snow at a wreck and then just stomp down on the binding and you are in. If seconds count (and sometimes they do), the faster I can get into my binders the better off I and the patient will be.
Skis: - turned up tails suck big time for the patients. We've all experienced this when skiing behind someone with twin tips at speed. The rooster tail if freakin annoying. When doing training, I got to ride in the sled a lot and I hated doing it with patrollers who had twin tips. The other problem I found with twins was if I needed to move back in the horns to control the sled. The tails sometimes get stuck/caught under the front of the sled and impede your ability to turn properly. On one powder day that was a training day, I wore my Praxis skis. A more experienced patroller, who had never ridden reverse camber, reverse sidecut skis in powder suggested they may not be the best for sled handling. While I agree with this on hard-pack, I think they're great on powder.
Snowboard? At the resort where I patrol we don't allow snowboarders to patrol. This is unfortunate, but I now understand better why we do it. There are too many areas of the mountain where you'd be slowed down appreciably if you had to respond on a board. Either hiking, traversing or skating, skis are going to be faster. Since you never know where on the mountain you might be when called to respond, I reluctantly agree that skis are a better tool for out mountain. Its too bad too, because I'm at least as good on a board as I am on skis and I'm certain that sled handling in nasty terrain would actually be better on a board - especially since I wear hard boots and ride alpine boards that give me incredible edge control and pressure in sketchy situations. One of the hardest things I found in sled handling on skis was side-slipping down in crappy conditions. Crappy could mean icy or it could mean lots of wet heavy slough. Either way it would be easier to handle on a board IMO. Skis are more visible from uphill when you cross them in the snow above a wreck though - FWIW.
Having knee pads is a nice thing. You spend a lot of time on your knees in some wrecks. If you noticed the neck trauma one last night, the girl was kneeling at the patient's head the whole time - which could be 20+ minutes depending on the situation.
As mentioned above, gloves get wrecked. We mostly use the Kinco 901 ski gloves, although I'm inclined to pick up a few pair of the BD Patrol gloves once I chew through my Kincos. They come up on SAC pretty cheap sometimes and they are way more comfy and dexterous than the Kincos. Pants also get wrecked. I cut slices in the thighs of a new pair of Arcteryx pants last year. Then I took them to a local seamstress/upholsterer and had her make me two heavy-duty cargo pockets on the thighs out of cordura balistic nylon. That worked great. Now I have two more pockets for a bloody kit and rubber gloves, plus it protects my pants.
Other equipment will probably vary by resort. We are required to carry a shovel/probe/beacon at all times (you can leave the shovel/probe at the top on very low avy-danger days if you want) so backpacks work better than fanny packs. I have a brand new Harper pack I bought as a trainee and never used if anyone wants one. My preference is a low-profile backpack so I can wear it on the chair. It holds all sorts of first aid gear, a spare hat (for patients if we take their helmet off), webbing, harness, snacks, dry gloves, etc.