Felling Axe
A felling axe is what most people think of when axes are brought into the discussion. They are typically made with lighter slimmer heads and meant to cut wood against the grain. These can be useful for felling trees or cleaning up branches. They are meant to be swung quickly and with little effort however they can easily get stuck.
Splitting Maul
Splitting mauls are basically small sledge hammers with a sharpened end. These are best for splitting wood as their name suggests. Swinging down at a log you can use the weight of the maul to provide the force to split with the grain. They have a wider head to assist with pushing the wood outwards increasing the splitting ability.
Hatchet
Hatchets are multi use tools, not the best at any one task other than being small and light making it easy to carry with you. They can be used for processing game, felling small trees, splitting kindling, or worst case scenario a hammer if need be.