Servos modified for full rotation lose position control. If you read about how a hobby servo works, and what the modification entails, you'll understand why this is.
The advantage over motors is that a $10 servo has control electronics/speed control (you don't need a separate h-bridge or motor driver), gear reduction, and an easy to mount to spline, all in a nice compact package.
There are the Dynamixel and Hovis lines of digital servos which can toggle between continuous rotation and position control. These are more expensive ($35 for the cheapest Hovis servo, iirc). The ~$200 Dynamixels have a ton of features, as well as 358° of position control. These servos also do not use PWM, and can instead be daisy-chained, which is awesome.
I can't think of any specific terminology, though...