An ax...Made to chop down trees with (as a logging tool)...Also used as a fireman's tool to chop into buildings with. Some axes are double-bladed, so that as one edge dulls, there is another sharp one in reserve.
A hatchet...A smaller version of an ax...(Generally used as a camping tool, and can be worn on a belt in a sheath.).
A tomahawk...A throwing/close combat weapon...Some/most with a spike on the other end of the cutting head...This was seen in use by American Indians, frontiersmen, and pirates (among others).
Come companies blur the line between a hatchet and a tomahawk, (such as ATC who call their ATC/Brend fighting (tomahawk) a "Fighting Hatchet"...).