Well.... as with almost anything, "that depends."

I live in Newcastle (Weston county), quite near Sundance (by Wyoming measure - 45 miles) and we have less than half the snow of a lot of places in Crook county. Newcastle is in a little "banana belt" and has far less snow and cold than most of the rest of the state.
But we still do get snow, ice and cold. It's all relative. Compared to Florida, it would seem extreme - at least at first. I came from S. Calif. and was sure I'd suffer terribly from the cold. But I didn't. I got used to it very quickly. That said, it does matter what kind of work you do. I'm retired, so if it is nasty outside - I just stay home. If you have to actually work outside all the time, it will be a lot more difficult.
The 4WD and chains question will be more related to where you live. If you are near mostly maintained roads, a 2WD may do, but if you are out of town far, a 4WD might be essential. You figure that out after you get here.

Never saw anyone here with chains on their car.
I've talked to a great many people about a possible move to Wyoming. Most of the folks from the south are more worried about snow and cold than anything else. You just have to figure what is most important to you. And make a visit in the winter so you really know what you are up against. All too often, our imagination of a danger or challenge is far worse than the reality.