Proper Use:
The double bit axe is usually used to fell, trim or prune trees and to split and cut wood. It is also used for notching and shaping logs and timbers. The single bit axe (flat head axe) in addition to the above uses, is used to drive wood stakes with the striking face.
Hatchets are used for cutting, splitting, trimming and hewing, and driving unhardened nails and stakes with the striking face.
Mis-Use/Abuse:
The cutting edges of axes and hatchets are designed for cutting wood and equally soft materials. They should never be struck against metal, stone or concrete. The striking faces of hatchets are properly hardened for driving common nails but should never be used to strike chisels, punches, rock drills or other hardened metal tools, or for striking stone or concrete. Never use an axe as a wedge or a maul. Never strike with the sides
Source: Hand Tools Institute, Hand Tool Use and Selection Chart