Like many choices, it kinda depends on what you plan to do with it. "Both" is actually a pretty good answer, because limiting yourself to either one would have you wishing for the other before too long.
Axe: if you are going to fell trees with an axe, and there are very many of them, or they're very big, you better be a hell of a man. Most woodrats carry an axe for tasks like knocking in wedges and to a lesser extent, limbing. For falling or bucking, a handsaw (bow or buck saw) is a better choice for crosscutting, and even that is a lot of work.
Hatchet: I use mine a lot in log building/timber framing, and I often carry it on regular jobsites instead of a hammer. I worked with a guy once who carried a hammer and a hatchet, and he frequently used the hatchet as a sort of "chisel with a handle", striking it with the hammer head. Anyway, for 90% of camping or woodsman chores, the hatchet is the way to go. Collecting up dead limbs for firewood, cutting shelter poles and stakes, opening vacuum packed jerky, etc. Lighter and shorter than the axe, you have it with you all the time.
Either one needs to be as sharp as you can get it, and the edge should be shaped for the most likely tasks. Keep the edge out of the dirt and protected from moisture.
Here's the big thing. The tool itself is not as big a factor in success as the skill of the user. Those GB tools are nice, my buddy bought some and they are fine tools. BUT. They cost a lot, and if you don't practice with them, you'll be holding them back, not the other way around. If you are already a skillful axeman, buy the GB's and don't look back, or have a pro blacksmith make you a custom axe. Since you're asking this question on this forum, though, I'm gonna suggest you find a used axe and hatchet for 10 bucks each or so. Stout hickory handles, not that tubular steel shiite. Look for ones with a thin edge angle and learn to sharpen them, then use them for a while. When you can handle them well, then you'll be in a better position to select a GB or Oxhead that's just what you want, not just pick one out at random. Or you can decide if you'd be better off to spend that money on a chainsaw.
HTH, best of luck.
Parker