Physicists have distinct meanings for the two. Mass can be described in two ways. It is an objects tendency to resist motion, or inertial mass. Something more massive tends to resist motion more than something less massive. Mass can also be described by its gravitational attraction to another mass.
Weight is the measurment of the force of gravity on a mass.
Weight can also be thought of as the force of gravity multiplied by the resistance to motion of an object, or intertial mass. Oddly enough, the weight of an object is the same in both cases. This is weird because there is no reason to believe that the intertial mass of an object should be the same as it's gravitational attraction, yet it is.