"Tannenbaum" just means pine tree, but that doesn't work very well in the song, now does it.

Christbaum (Switzerland); Weihnachtsbaum (Austria, Germany)

Tannenbaum - in the famous song refers to the actual growing tree
Christbaum und Weihnachtsbaum are the same, its regional preference which one to use.
Christbaum is more in the southern part, Weihnachtsbaum preferred in the northern parts of Germany.

HTH
Sabine