couldn't I just cut all the other branches off that tree?
And waste a perfectly good tree?
Actually, I carry both a saw and an ax. A saw is good for cutting small branches and trees, but if you have to split something, then you'll need an ax. I should also point out that not all saws are created equal. Some cut slower than others. In the right hands though, an ax can clear branches from a fallen tree much faster than most saws, including chain saws. Then you have the added utility of having a big hammer handy.
I should also point out that there are different types of axes for different uses. A splitting ax doesn't work as well at felling trees, for instance, as a felling ax, and vice versa. Most axes you'll seen carried tend to be of the splitting variety, which is where they come in handy. For the rest of the time, a hand saw will do whats its designed to do and cut while the ax splits.
And as mentioned above, an ax makes a very handy extraction tool in case of an accident. Lets see your bow saw do that.