Momentum is mass x velocity. Kinetic energy is mass x velocity squared.
Energy is what drives all of it. As Marshy said force is what splits the fibers, and many things determine how effectively the energy is converted into force - face angle, etc. The inability to understand energy flows and basic physics comes up again and again.
Force is expressed in pounds, a wood splitter has tons of force but zero initial velocity and therefore zero momentum. I maintain that when we manually split wood that momentum is required and that kinetic energy is irrelevant. A bullet is not the best example because it is not an inclined plane but an arrow is
A hydraulic wood splitter is not the same model - none of us is strong enough to push a maul through the wood by applying a continuous force. Instead we must store energy in the moving mass of the maul head and transfer it to the round where it can be converted into forces that separate the wood fibers. There are splitters that work off of impact like we swing a maul - what are those called again? Oh yeah: Kinetic Energy splitters. Splitting with an axe or maul is a discontinuous process where the energy is transferred in separate events, and at each event you only have the amount of energy you transferred to work with.
The amount of energy you transfer to the wood is determined by KE = m x v^2, which is a magnitude, not a vector - again, it's just how much energy you have to work with.
Force is a vector which can be calculated by F = m x a. Acceleration is the change in velocity over time, so if the maul slows down slowly then the acceleration was less, and less force was applied - but all the energy was still dissipated. Force is also Work/displacement (work is a measure of energy, and in this case is the KE you supplied). So if the splitting tool moves a long way when it hits (as a sharp axe will sometimes), then the force developed was less. Both of these are the same concept - of the tool stops fast a higher force was developed by the energy you transferred to the round.
But force is still a vector quantity and nothing says that the force was properly directed to split the round effectively.
The momentum transferred to an object by a force equals the force times the time the force is applied (Momentum = force x distance). Force comes from energy, so that can be converted into: Momentum = Energy/distance x time. It's still limited by how much energy you transferred to the round.
Energy is what you have to work with - it could all be turned into heat without accomplishing anything (like if you swung a sledge hammer instead of a maul), or it could be effectively converted to appropriate forces and split the round. Or maybe it's not enough energy and the round does not split. Or maybe the round is on soft ground and the force is converted into momentum and transferred into the ground. Lots of variables.
Still, if you transfer more energy you've got a better chance of success, and that means you want to increase the velocity, because in all these equations there is only one squared term.