A hawk has a long, knifelike blade that enlarges rapidly into a large eye. This allows you to slide the handle on and off easily, and is easy to improvise a haft in the field. A hawk head is lighter, which makes it easy to maneuver as a weapon. The large eye makes it a less efficient wood splitting tool, because if you try to use it as an axe, the hawk will not penetrate past the head unless you baton it. I suspect the knife like bit is less efficient at chopping too, especially as the diameter of the wood gets bigger.
A hatchet is a more modern evolution of the hawk, with its wedge shaped head. It has very gradual transitions from blade to eye, and pushes through wood more efficiently. Also, the head geometry concentrates more weight at the head, which makes it bite deeper. If you wanted to process lots of wood, axe is the way to go. I was able to section and split wrist sized wood with no trouble using a hawk, but for bigger stuff, I'd have to start using wedges to avoid wasting energy.
While an axe is an efficient tool, my heart goes to the hawk. After all, it was the tool of the frontier, wielded by mountain men and indigenous people alike. I suppose if it served them well, it's good enough for me. I'm still learning how to best use it. For example, I think to split a log with a hawk, it is better to lean it on another log and hit up and down its length until it splits. It may take 3 blows with the hawk and 1 with an axe.
I will use the hawk whenever possible. However, if someone said, "you have to survive a week in the woods in winter," I'm taking an axe.