What are the advantages of 10 speed over 21 speed bikes?

Depends a bit on which year you're talking about. Way back 10-spd used to mean two chainwheels at the front and 5 sprockets at the rear, making it a total of 10 speed. Today 10 speed usually means that you have 10 sprockets at the rear, and then 2 or 3 chainwheels at the front, making it a total of 20 or 30 gears. So if you have an old 10-spd(2x5) then the less old 21(3x7) will offer a greater number of different gear ratiOS. But if you're comparing a modern 10 speed(2-3x10)with a somewhat old 21 speed(3x7) then the 10 speed is likely to be a newer and better bike.