that's a tough one, brother. good question.
i think most folks don't really know, but (to me) the thing that really differentiates an axe from a hatchet is the head geometry, which is moderated by two factors predominantly;
an axe head can be very specialized in cross section, and i think that is a good practice the more i chop with one - since they are heavy comparitively, it stands to reason that you are going to be doing something more-specialized with the axe on large scales, while the specialized things that you might be doing with a hatchet will be small scale (carving, for example - Ishi loved to shape his bows with his hatchet, for instance), in optimized and ideal settings at least. - that's lot of theory there in one sentence, isn't it.
hatchets will also prevalently be straight-bit'd, which can be nice if you care to try carving with them.
these are just my opinions on the matter, definitely not gospel.
nothing better than a good carving hatchet, except maybe a good adze.
summary.
1) an axe can be almost anything in cross-section, as it is optimized for specific types of tasks, while a true hatchet will tend to be wedge-like in cross-section, so that it can split fairly well, while still being a decent chopper and limber by virtue of its light weight assisting its poor cross-sectional geometry, when compared to its big brother, a proper axe.
2) an axe tends to have a more-curved bit, while a hatchet will often have a larger radiussed bit to perhaps something completely straight-edged.
...so essentially, a hatchet tends to vary by its edge geometry and remain a hatchet, while a fine axe tends to vary in its cross-sectional geometry, while remaining true to its respective speciality; large, fast work.
man, i didn't realize what a complete chopper geek i was until today.
hatchets are great basecamp tools.
HTH.
vec