A slave is someone who is bound to another -- his or her master. The master chooses the slave and usually purchases the slave with a price. The slave does not choose his master -- he is not free to choose another master. The slave must do whatever his master requires of him -- he is not free to do as he wishes, unless so granted by his master.
A saint is someone consecrated to God, to serve Him. When we confess our sins and acknowledge Christ's death on a cross to pay the penalty for our sin and invite Him into our lives, we become consecrated to His service. We no longer live for ourselves but we live for Christ.
Both are much the same as neither is living for self but living to serve Christ. I think there is a difference of degree, however, in that Paul was specifically chosen by God for a special service and he feels he is compelled to perform this service every waking moment. In his case he feels he literally has no freedom whatsoever. We as saints, however, have a great deal of free will -- we serve Christ, but not by the same degree of dedication.