A quick correction first: don't use "of" in the phrase "I'm not against." You just say the thing that you don't oppose right after "against." Example: "I'm not against the war." To your question, they both have similar meanings: you don't oppose something; you won't try to stop it. There are a few differences though: 1) "I don't mind" is much more common in conversation and is less formal. If someone says, "I'm cold. Can I close the window?" you should answer, "I don't mind." No one would say, "I'm not against your closing the window." 2) "I don't mind" has the meaning of "It doesn't bother/irritate me", but "I'm not against" doesn't. Example: "I don't mind my little brother, even when he's crying" means that he doesn't irritate you. "I'm not against my little brother" means that you don't oppose him in a competition or contest, in a game for example. ("I'm not against my little brother in the soccer tournament. He's on the same team as me.") Less commonly, "I'm not against my little brother" could also mean that you don't oppose his existence; you don't want him to stop existing. Another example: "Donald Trump isn't really against North Korea. He just wants to make his supporters happy." This means that Trump doesn't really want North Korea to stop existing.