Sunstone said it in the OP.
Well Obama himself doesn't speak against mixing religion and politics. What he said is that politics should not be sectarian (promoting one sect over another) and that we shouldn't use faith as a means to attack people, and
most importantly that we should be willing and able to translate our faith-based positions in terms that people who don't share our faith can understand. That last part would make no sense if he was actually saying that you can't mix politics and religion.
Church and State. The mixing of Church and State would be any time the State (ie - govt) does something to either promote one religion above others. For example, if it made Christianity the official religion of the U.S. OR the govt does something to prevent someone from practicing his or her religion. For example, it makes praying illegal. Both of these kinds of activities are prohibited by the First Amendment of the Constitution. Thus, we have the separation of Church and State.
Religion and politics. Someone believes because of their religious convictions that abortion is wrong. Because of that, they vote for the candidate who says they're against abortion. That's the mixing of religion and politics. OR someone believes of their religious convictions that all people are created equal. Because of that, they lobby their Congressman to pass a law granting equal marriage rights to BGLT folk. That's the mixing of religion and politics.
What Obama was saying in his speech is that if you want to say that abortion is wrong, you can't say "Because the bible says so." That would be imposing sectarianism - claiming that belief in the bible should hold sway over everyone. (It would also be wrong because the bible doesn't say it.) But you can believe that abortion is wrong based on your religious convictions and argue against abortion based on other lines of reasoning that would be understandable by everyone.