"Preaching to the converted" is essentially what it says, ie trying to justify something to people who already share your opinion, so the idiom is to express the point that there is no actual need to say anything.
"Preaching to the choir" is the United States version, which means exactly the same thing, and which several English people (ie Richard Dawkins) use, since his predominant target audience are Christians in the US (the UK is already quite an Atheist country) so he uses the American idiom.
As is common with American versions of English idioms, the American version does not actually make sense, since preaching to anyone in a church would suggest that they are already believers, so "preaching to the choir" would suggest that preaching to the congregation, or to the choir would somehow be different and it is this apparent ambiguity which is causing so much of the confusion in the above answers.
The English version is quite obvious as "the converted" would be people that you would not need to preach to. Much of the confusion in many of the answers seems to stem from this basic lack of clarity in the American version. The intention and meaning of both is identical - there are no subtle differences as people seem to be stating - but the American version just contains ambiguities and is just less clear.
Another example of such a poor American idiom is to say eg "I could care less" when the English idiom is "I couldn't care less", which actually means that I do not care at all, hence I couldn't care less. The American idiom which is supposed to mean the same, actually means the complete oposite, since if one could care less, then one must care some non-zero amount. In fact one could care a great deal and could still say "I could care less", so in fact the American idiom actually means the complete opposite of what is actually said.