Gamertag is your unique identity on the Xbox Live service. It represents you as a unique individual amongst millions of other Xbox Live users. An Xbox Live Gamertag can be up to twelve characters long, including numbers, spaces, and letters.
Gamerscore is an achievement counter that reflects the number of accomplishments gained by a user on the Xbox Live platform. Every game comes with a certain amount of points a user can achieve by completing its challenges, tasks, and other inputs. Gamerscore is available on every Gamercard (an information panel that includes Gamertag, recent games played, reputation, avatar, etc.).