Good lord, you guys are trying to get me going, aren't you!!!
Here are some points (some have already been made):
1) Fiskars :barf: Enough said.
2) Anyone can pick up a saw and use it good. Not much training there. The same is not true of the axe. Many spend their whole lives dedicated to axemanship, and still don't know it all. There is tons of technique and experience involved. I would be surprised if you could just pick one up and wield it like an expert.
3) Like mentioned, the saw is good for one thing. An axe can chop, limb, split, shave, skin and cut.
4) If you bend or break your saw in the field, you are stuck. Field sharpening is difficult at best. Usually handle issues with an axe come with plenty of warning, if you know your axe and are paying attention. Aside from the handle, it is pretty difficult to damage the head beyond a "useful" state in the woods.
5) I think bushcrafters in the UK prefer saws because they do not have the vast areas of wild places we do, nor the resources left. Plus, they are forced to be a much more PC lot than we are

Poor them! I hear they are also talking about taking points off their knives over there. No joke!
As mentioned is Mors' "Bushcraft" a saw is never a replacement for an axe. It is rather to compliment it. If you have to chose just one, the axe wins every time.
Here is the real thing to think about:
Do your outdoor activities really rely on your having a saw or an axe? I would assume the answer for most is "no." That is probably why this debate keeps surfacing. We have all sorts of gadgets and gear, and do not necessarily need to craft the things we need in the woods anymore. Unless, like me, that is what you enjoy and appreciate, and also like developing your axe skills.
B