Heck, Ive got a big mouth, so Ill hop in on the conversation.
Jeff and Hoohoo show some great insight on the subject, especially since they spend time in entirely different climates. Im a northwoods man myself, much like Hoodoo, so I use hatchets and axes.
If I were to car camp, though I dont, Id have all manner of axes, hatchets and saws with me. If I were to go backpacking for any reasonably long distance, Id probably just take a SAK and a good sturdy fixed blade. The trails I walk have plenty of wood on the ground for a good fire. If I run into a serious obstacle, I walk around it.
If a northeastern trail doesnt have much wood on the ground, the reason probably has to do with overuse by campers and hikers. In such circumstances, fires should not be built. You end up with a forest floor filled with pine needle and such, which is a real fire hazard. Any more thinning of the understory would only make matters worse.
I do agree with Hoodoo, after a bit of thought, that an axe would be very appropriate for any cold weather travel.