"Band-mate" is a term that one member of the band might use in speaking of another member of the band, or that one might use to emphasize the relationship between two people who are members of the same band.
"Joe is my bandmate" said Fred, "and has been since we both started performing."
"Judy and Chris remained band-mates through the formation and breakup of three different groups.
"Member" is the more general word and is far more common. Bandmate always emphasizes the relationship between two or more participants in the band.
Other terms such as "performer", "participant", and descriptions of a specifc role such as "lead singer", "bass player" and the like can also be used when appropriate, but member is probably the most common usage.