The answer to this depends to some extent on what country we are talking about.
In Great Britain, the anti-slavery movment accomplished two major things. First, it got the government to ban slavery in Britain and its colonies. Perhaps just as importantly, it got the government to ban the slave trade and to use the Royal Navy to enforce this ban. This greatly reduced the number of Africans being taken as slaves.
In the United States, the anti-slavery movement helped to bring about the Civil War. This war, of course, led to the freeing of the slaves in the American South. Emancipation of the slaves was the major accomplishment of this movement.
The anti-slavery movement just abolished the right to own slaves in the south. They didn't eliminate discrimination or abolish slavery in the north.