Actually, what Luther said was “sin boldly, but believe even more boldly.” What people do not understand is that for Luther faith and love always go together. So in a sense he was saying something quite similar to Augustine’s dictum. He was not actually recommending sin or saying that sin was OK–he was saying that we should not worry about sin but should believe in Christ and the rest will follow naturally. (This is obvious if you take the trouble to read Luther’s theological writings instead of basing your entire view of him on one out-of-context letter.)
The big difference between Luther and Augustine was precisely that Luther thought faith was itself the means by which we receive grace, and love was not something that needed to be added to faith. Furthermore, from Augustine’s (later) perspective both faith and love (and the resulting good works) were the result of God’s grace, so that a distinction between faith and works wasn’t that important. Luther disagreed. For a thousand years Western Catholics had been repeating Augustine’s dictum that when God rewards our merits, He is crowning His own gifts. Luther didn’t think this was good enough. He thought that even seeing our actions as the fruit of God’s grace working in us did not prevent us from taking pride in those actions. All saving efficacy had to be denied to good works, period. In this (and in the doctrine of “forensic justification” by which he accomplished this goal) he was very different from Augustine, and he and Melanchthon both admitted this in candid moments!