A gransfors hunter's axe goes everywhere I go. I always have at least one blade with me but if I had to chose between the two I would take the axe over any knife.
The axe is easy to carry on a pack or in the middle of my bedroll where it helps keep everything in place.
Pretty much what he said.
I'm a big fan of the GB hunter's axe. If I was going to bring one sharp tool, it'd either be this, (or if I went the end of the world route, or just a long camping trip, or in the dead of winter, like now) or the GB Scandinavian Forest Axe.
The 19" Hunter's or Small forest are easy to carry, especially if it's on the pack and not the belt.
Personally, I'm better with the axe than with a knife. I can make fuzz sticks easier, I can butcher game easier, and I found out last weekend, I can carve a spoon easier with the Hunter's axe than with a knife (and for the spoon it was up against a dedicated 2" long carving knife).
At the risk of stepping on toes, the idea that axes are somehow more dangerous is simply foolishness. I have to assume that it's being said by guys that just don't know how to use one or are repeating something they heard. There are a few simple rules on using an axe. If you follow those rules, they are perfectly safe. The videos posted in post #15 go through most of what you need to know. Books like Mors Kochanski's Northern Bushcraft or even old Boy Scout Fieldbooks go over it as well.
Many don't seem to know it, but you can baton an axe as well. Pound the head into the end of the log, and then use the baton along the axis of the axe. Keep the haft at an angle so the baton doesn't hit your hand. There's also easy ways to split smaller wood -- again see post #15.
Maybe it's just me, but I can go through wood much faster with my Hunter's Axe of Scandinavian than I can with a saw, unless it's a chainsaw or two-man felling or buck saw, on very large (multiple feet in diameter). Only time I use a saw is if I want a straight cut for making something.
The only things axes aren't good for is use as a draw knife, cutting light vegetation (though they can do it) and for those that simply cannot fathom that an axe is safe to use when splitting wood. In all those situations, a machete absolutely
DOMINATES over a large knife or saw.
Blitzkrieg -- For you, and your end of the world thing specifically, here's where the axe rules over the saw: What do you do in your extended live in the woods for years on end scenario when your last saw blade breaks or can't be sharpened?
Referencing the above, I'd say that a 3/4 axe (24-25" long, 2-2.5 pounds) is worth its weight, because you can use it FASTER to build shelters and get wood. However, a 19" axe like the GB Hunter's or Small Forest will do. If you can't carry the weight, I have doubts about your long term survivability in the woods.
Take one of those axes, an 18" machete, and a smaller (3-4") knife, and you can rule the world.