Not very much, says Alex Tabarrok:

Why are more independent central banks better at fighting inflation than less independent central banks?  There is nothing magical about independence that makes for low-inflation.  Suppose we pick someone at random and give them complete power over monetary policy.  Such a central banker would be very independent but I wouldn't count on this policy resulting in much in the way of systematically lower inflation.

The primary reason that independent central banks are better at controlling inflation is that absent direct political control the default selection mechanism favors bankers, i.e. lenders, people whose interests make them more favorable towards lower inflation.

Thus, independence is a political decision that favors lenders in the decisions of monetary policy.  Now, depending on the alternatives, there may be good reasons for making this choice but we should not fool ourselves into thinking that we have depoliticized money.  We should not be surprised, for example, that "independent" central banks tend to make lender of last resort decisions that protect banks and bankers.

I don't think this is quite right. 

Start with a stylized fact:  in a democracy, there will be a strong tendency for monetary policy to favor debtors, because there generally more debtors than creditors.  This is particularly true of America, with its lavish credit markets.

In the long run, however, strongly inflationary monetary policy makes everyone worse off; it impedes capital formation, lowering productivity.

Central bank independence works, not because the bankers aren't accountable to Congress (they are, after all, reappointed every so often), but because Congress is only weakly accountable for the actions of the central bank.  If Congress were held to account for the actions of the central bank, Congress would appoint bankers who would do populist things that would make us all worse off.  Most of the financial policy journalists I know have the sense that Congress actually supported most of what Paulson/Bernanke/Geithner did, but knew they did not dare enact it.  They don't want a more accountable central bank.

So there is something magical about bank independence.  It lets Congress cut against its basically populist political interests.  You may think that makes it too bank-friendly.  But it also means we don't have double-digit inflation, which is where we were headed before Volcker.