By common definition a slave is a person owned by someone else, the "master", and is obligated to serving the master's request no matter what the outcome. The definition of the word used by Paul is to refer to a person who has chosen to be obligated for life by a desire to follow and obey Christ Jesus.
The root idea of a saint is one who is sanctified, or set apart, dedicated to serving God out of desire and obedience to Jesus.
In the sense of my relationship to God, they both indicate a willing desire to follow and obey the Christ, and to be committed to that service for the rest of my life regardless of obstacles or difficulties I encounter on the journey.