Cutting axes are very thin as they are used to penetrate across grain. This thinness means that it presents max force on small area similar to a razor blade in this respect. This causes it to sever instead of wedge which makes it less useful for splitting but great at cutting.
A splitting axe has a wide angle blade and when slammed onto a log it has a wedge like action and forces the log apart. It expends all its energy in this way and therefore causes a deep split that forces the grain apart. Because of this it cannot be used to cut across grain as it will not penetrate nearly deep enough and will lead to a shallow cut.
Curved or Straight bit
A curved bit means only a small point of the wood will be in contact with the axe at a time and this aids in wood penetration either with or across the grain. A flat bit gives an even cut rather than a deep cut and there are varieties on this theme. The broad axe has a flat bit and a ramp on one side. This allows it to cut out a very flat section of wood but for general purposes a rounded bit is much more versatile.
Single or double bit
The single bit axe is more common and has certain advantages such as its design is optimised to put all the torque behind the one bit it has. A double bit axe offers different advantages and is generally symmetrical around the center line. This does offer a certain amount of balance that isn't found in other single bit axes. The two blades are often ground differently to offer both splitting and cutting prowesses and therefore grants a large amount of versatility however these axes cannot be bludgeoned.