The Concorde and the Tupolev both relied on afterburners, because they operated under similar design constraints -- the "western" jet engines in the Concorde were not that much better than what Soviet design bureaus could produce.

The Concorde was much smaller, and lacked one of the major innovations of the Tu-144 -- forward flap canards to improve handling on a larger jet.

Probably for the better. The Tupolev killed a lot of its passengers, and it was almost immediately withdrawn from service after the first few incidents. The Concorde, a simpler and smaller design, served for decades.