Apostle: One of the twelve disciples chosen by Christ to preach his gospel. The word comes from the Greek word meaning messenger.3
As can be seen, there is a difference between disciples and apostles, in that anyone who follows a certain person – in this case Jesus Christ – can be called a disciple, whereas an apostle is a person who has been specifically called and set apart to serve as a messenger or witness for Jesus Christ. Apostles are also disciples, since they are followers of Christ, but disciples are not necessarily apostles.
This is not to say that Christ’s disciples are unimportant; quite the opposite is true, for these are the people who take on the other roles essential to the Church’s structure. In the LDS church, these disciples take on the roles of pastors (branch presidencies or ward bishops), teachers in the various auxiliaries (primary, relief society, Sunday School, young men and young women and so on), evangelists (aka seventies4 and missionaries).
1 Articles of Faith, 6; Ephesians 4 (see v. 11 in particular)
2 Reverso Dictionary disciple, definition
3 ibid. apostle, definition
4 Luke 10: 1-20
The painting on the left: The Sermon on the Mount, by Harry Anderson
The painting on the right: The Last Supper