What's the difference between a covenant and a contract?

While they are equivalent in many contexts, there is an important nuance to covenant. In the words of philosopher Douglas Groothuis, "A covenant, in the theological sense, is more than a contract. A covenant is made with a sense of honor and obligation before a transcendent reality."

Marriage is a contract—but more, as vows are made in the sight of God and in line with his purposes. Thus it is also a covenant. Some wrongly think that covenants are unbreakable, unlike contracts. I believe this is not correct. Both contracts and covenants are binding—which is why violation is a serious matter.

To illustrate the feel of covenant, consider the Covenant Prayer of John Wesley (1703-1791):

I am no longer my own, but yours. 
Put me to what you will, place me with whom you will.
Put me do doing, put me to suffering.
Let me be put to work for you or set aside for you—
Praised for your or criticized for you.
Let me be full, let me be empty.
Let me have all things, let me have nothing.
I freely and fully surrender all things to your glory and service.
And now, O wonderful and holy God,
Creator, Redeemer, and Sustainer,
You are mine, and I am yours. 
So be it.
And the covenant which I have made on earth,
Let is also be made in heaven, Amen.

Imagine how differently this would feel if expressed in the words of a cold and clinical contract: "Since God takes care of me, I will obediently do he says. If either party fails to comply, obligations are null and void"—or something like that.