IMO, Black Hand pretty much nailed it (sorry for the pun). A saw can cut wood across the grain, and that is about it. The others can all cut, scrape, chop, split, etc, etc.
I'm a blade freak. When I go backpacking or camping, I usually carry a folding knife, like CRKT M16, in my pocket, a fixed blade knife in a belt sheath, such as the Gerber Strongarm or Cold Steel SRK, and small hatchet of belt axe strapped to the pack. I can always be sure having at least one blade with me at any time.