The mountains in the West are larger, higher elevation, and therefore possibly more strenuous than what you're used to. So: extra physical preparation and altitude climatization.
Generally the Western mountains are drier than the East, so you don't need as much of the equipment for rain than you would out East (unless you're backpacking in western Washington state).
animal-wise, it's about the same except out here there are Elk, which in their behavior and their interaction with humans (or lack thereof) are basically "large deer" meaning there's nothing special you need to do. I don't know whether moose live in New England, but depending on where you are in the West, you may find yourself in moose country and this is more of a concern as moose can be very dangerous.
Stream crossings. I somehow picture streams in New England as placid, gentle-flowing things. I don't know whether that's true or not, but out here in the West, mountain streams and rivers are often very swift and dangerous to cross. Even small ones. So you need to be aware of the dangers of stream crossings. In my life, there have been a couple of very precarious and life-threatening situations I found myself in while fly-fishing streams in the Rockies. So you need to be careful.