According to Reformed Theology, how is Compatibilism different than Hard Determinism?

I mentioned it in the comments, but I might as well post an answer. As @bradimus brought up though, this might be better for the philosophy QA site.

Compatibilism does not deny free will. Although, as the wikipedia page says:

Compatibilists are sometimes called "soft determinists" pejoratively

Reformed theology doesn't have anything to do with this - reformed theology denies free will and do not adhere to Compatibilism. And by free will I simply mean total autonomy and agency - ie, when this article says:

has it in his power to either do good or evil toward God

As an side - I am not necessarily reformed (but I think lots of their ideas have biblical and logical merit), and I would identify as a Compatibilist. I think God basically does what He wants, as He is Lord - which I can prove by citing.. well.. the entire Bible I guess.

I don't think this conflicts with His gift of agency to us though. Sometimes God 'hardens hearts' - and we have to deal with that. But at the same time, so often we pray for guidance on things, and He doesn't seem to give us answers, perhaps because either way is fine with God.

Or perhaps He is more interested in what we decide on our own. There are precedents for both in Scripture. (Pharaoh in Moses' time, Saul choosing to disobey God, David's adultery and subsequent repentance, Paul/Barnabas split and their ministry decisions, Peter's Vision and subsequent teaching/lifestyle changes... etc. etc.)

Definitions:

• Libertarianism= everything free will.
• Hard determinism= no free will at all.
• Compatibilism combines these and says they are compatible with each other.

Your question assumes that Reformed Theology has a view of compatibilism. My reason for agreeing that it does is that A.W. Pink, W. Grudem, D.A. Carson, J. MacArthur and J.I. Packer for example would all agree with D.A. Carson's explanation of compatibilism: I'm sure of this but I only mention their names so that others can check out the details for themselves. If I included quotes from all of these then this answer would be over long, I guess.

Carson:

1. God is absolutely sovereign, but his sovereignty never functions in such a way that human responsibility [ and freedom] is curtailed, minimized,or maligned.
2. Human beings are morally responsible creatures- they significantly choose, rebel, obey, believe, defy, make decisions, and so forth, and they are rightly held accountable for such actions; but this characteristic never functions so as to make God absolutely contingent.

A. W. Pink in "The sovereignty of God" puts that more simply [P9], "God is sovereign, man is responsible".Pink continues on page 9 to say:" To emphasize the sovereignty of God, without also maintaining the accountability of the creature, tends to fatalism".

"Fatalism" then, is something Pink is trying to avoid. I am confident that Pink's "Fatalism" just about = your "hard determinism". Let's try and look at- Reasons against Fatalism. It just reduces man to being a puppet. We are free to choose. [ But what if we are puppets and God holds all the strings, doesn't John 3v35 say, all things are in His hand]. Hard determinism is, man is what he is, sinful, fallen, in need of redemption if he is going to go to heaven, but everything that occurs is dictated by God and man's fate depends upon God's will alone. Can puppets love God? "Yes they can if that is the way their strings are pulled. Or, in Bible terms, if the Holy Spirit chooses to come upon them to make them born from above". This is, I think, the reply of the hard determinist. ["born from above" is literal translation of "anothen" John3 v7 and emphasises initiative taken in heaven rather than "born again" which might be result of human initiative]. Why translate it as born again unless one is trying to keep compatibilism alive? That last sentence is my own comment but the fact that I made it at all shows how important this question is.

In reformed theology compatibilism reigns and hard determinism is not usually thought about. But that does not mean that hard determinism has nothing to say. "The lives we give back, apodo 1 Pet 4v5, an account of, are first given to us. We first give to God nothing, for from Him are all things Rom 11v35/36", is as I understand it, the cry of the hard determinist.