Compatibilism is sometimes called soft determinism. It is distinguished from hard determinism. William Lane Craig, J.P. Moreland, and other libertarians sometimes argue that compatibilism reduces to hard determinism, and there really is no meaningful difference. I don't remember if it was Craig or Moreland (I think it was Moreland), but one of them compared compatibilism to a row of white dominoes where one domino was red. The red domino was supposed to represent the will--the faculty of volition. Since the will is causally determined by prior factors, such as motive, desire, belief, etc., then it's just one domino in a strictly deterministic causal chain. The redness of the domino makes no difference.
I think the major (and quite obvious) difference between compatibilism and hard determinism is that in hard determinism, the causal chain is made up entirely of non-rational, blind, mechanistic cause and effect, but in compatibilism, there is a mental component to the causal chain that includes a person's own desires, motives, inclinations, and intentions. And that is not an irrelevant difference.
The problem with reducing compatibilism to hard determinism is that if naturalism were true then compatibilism could not be true, but hard determinism could. The reason is that in compatibilism, mental events are able to have causal influence in the physical world. Things like motives and desires are mental events that have causal influence over the brain. If naturalism is true, then hard determinism would be true, but not compatibilism.
As Moreland often likes to point out, the indiscernibility of identicals dictates that if there is any property A and B do not have in common, then A and B cannot be the same thing. Since I've pointed out a rather obvious difference between hard determinism and compatibilism, it follows that they cannot be the same thing.
A person might respond by saying that even though there's a difference between a white domino and a red domino, it is not a significant difference. But significant with respect to what? I don't remember all the relevant differences Craig and Moreland pointed to, but there are three I can think of--an argument for the soul, rationality, morality.
An argument for the soul
In Scaling the Secular City, Moreland made the following argument:
1. If all we are is the sum of our physical parts, then we cannot have free will.
2. We do have free will.
3. Therefore, we are not merely the sum of our physical parts.
The reason Moreland used free will as an argument against physicalism is because physicalism entails determinism. But if we have a soul that stands outside of the physical world, but has causal influence over it, then it's possible to have free will. In fact, that's the only way we could have free will.
But you don't need free will to make this argument work. All you need is for mental events to have causal influence over the brain, which is possible under both libertarian free will and compatibilism.
1. If all we are is the sum of our physical parts, then mental events cannot have causal influence over physical events.
2. Mental events can have causal influence over physical events.
3. Therefore, we are not merely the sum of our physical parts.
Rationality
Moreland has argued that free will is necessary for rationality. But in other places he has argued that our beliefs are not under the control of the will. I asked him about that on the Alaska cruise, and he explained that although our beliefs are not under the direct control of the will, we do nevertheless choose to follow arguments, choose to think thoughts, choose to expose ourselves to belief-changing situations, etc., and our beliefs can change as a result. And that's what makes us rational.
But this account doesn't explain why it is necessary to have libertarian freedom in order to be rational. It seems to me that if my choice to follow an argument and think about its premises is the result of my desire to arrive at the truth, then I am being rational. And if my belief is the result of "seeing" the logical relation between statements such as "All men are mortal" and "Socrates is a man," then it seems to me that I'm being rational.
Morality
Craig, Moreland, and most (all?) libertarians think libertarian free will is necessary for morality and compatibilism or determinism is inconsistent with morality. I've already argued otherwise in my series on "An Argument Against Morality From Determinism," in May of 2005, but I'll give a brief response here.
According to libertarians, if our choices are determined by our desires and motives, then we can be worthy of neither praise or blame. If that's true, then the stronger our motives and desires are, the less culpable we are. The reason is because the stronger our motives and desires are, the more difficult it is to resist giving into them, and the more difficult it is to resist giving into them, the closer they are to determining our choices. It follows from this point of view, that the more full of hate and malice a person is, the less blameworthy they are. And the more full of love and kindness a person is, the less worthy of praise they are.
That's what follows from the libertarian view, but the libertarians have it entirely backwards. We all recognize intuitively that being a rotten person consists in having rotten desires, and the more rotten a person's desires, they more wicked they are for acting on them, and the more we blame them. And the more kind a person is, the more we praise them. An act that arises spontaneously for no reason at all can be worthy of neither praise nor blame, but we praise people or blame them because they do exactly what their wanted to do. The less hand their own desires have in bringing about their behavior, the less control we say they have over their actions. But the more hand their own desires have in bringing about their behavior, the more control we say they have over their actions. It follows that people are the most culpable for their actions when their actions are completely determined by their desires. Compatibilism makes sense out of moral responsibility. Libertarianism doesn't.