IMHO, a saw is safer than an axe and a better cutter for the weight. A Gerber Sportsman's Saw is a little over 3 ounces and a Kershaw/Bahco is a little over 6oz. IIRC, the lightest axes are around a pound. Of course an axe is better at splitting and the small ones can be batoned for splitting where they don't have the weight to carry through. My preference is to a saw. If I were in bad shape, I think the chances of whacking myself in the shin with an axe are too high, and that sure as heck wouldn't improve my situation. I lie a saw for cutting shelter poles and such. You can certainly do the same with an axe.
If you have skills you want to use or practice that require one tool over another, then there you are-- take that one. All-in-all, if you go out in the woods and you have a decent knife, and either an axe or a saw, your backside is pretty well covered for cuttin' tools. Heheh-- no one will acuse you of being under-equipped!
If you have a group on a hike, I think the way to fly is to have one person take a saw, another an axe, etc. Teamwork!