Sorry, I meant between other churches in the LDS movement and the main one.
Oh, that's easy; there's the Fundamentalist LDS church (FLDS), who as you noted, still have polygamy. They also still don't allow blacks to have the priesthood, and IIRC, they have had problems with hate crimes against blacks. Occasionally, they get one-man off-shoots of themselves, self-declared "One Mighty and Strong" sort of fellows, like the Elizabeth Smart kidnapper.
Back in the past, a few of the split-offs didn't agree with polygamy (or denying blacks the priesthood), and one of them became the Community of Christ group (also known as the RLDS). Community of Christ even has women priests, if I recall correctly.
The Ridgonite/later Bickerton split off has a president, but not prophet of their church. I don't know much about them, but they apparently believe that Jesus and God are one person, unlike the main church. They also don't believe in the Holy Ghost. They don't believe in the Doctrine and Covenants and the Pearl of Great Price, but they DO believe in the Book of Mormon. They do feet washing as well, which the main church doesn't do.
Oooh, and there's the
Strangites. Lead by James Strang, who was a complete nutcase. Made his own ancient records (and I think that they're still around, so you can tell that they're his own), declared himself King of his church, and essentially lorded over Beaver Island. Strang later was shot by members of his own Church because he was being a tyrannical loon.
All of these groups are still around today, but the biggest is Community of Christ, about 250,000 members strong. Next biggest is the Bickerton offshoot with about 12,000 members. The FLDS have about 6-10,000 members, with the number fluctuating through the years. The Strangites are the smallest, according to Wikipedia, they've got about 50-300 people.