Splitting and chopping wood. An axe is a two-handed tool for felling and splitting rounds; a hatchet is the one-handed version for kindling, limbing and camp work. Both cut by driving a wedge through wood with momentum, which is a completely different action from a knife.
Size and hands. A full axe has a long handle for two-handed swings and enough head weight to split rounds and fell trees. A hatchet is short, one-handed, and suited to kindling, limbing and camp chores.
See can a hatchet replace an axe.
Both use a convex edge, which puts the maximum amount of steel behind the apex and acts as a wedge to force wood apart as it enters. That is why they are ground convex rather than flat.
They are also run soft, so the edge deforms instead of chipping on a knot or a stone.
Split on a chopping block, never on the ground and never with a hand steadying the target within the arc. Clear the area behind you as well as in front.
Sharpen with a file and finish on a strop or with sandpaper on a compliant backing. See sharpening a convex edge.