What is the difference between stepper motors and servomotors?
There is a lot of noise in the answers to this question that seem to be conflating "servo motor" as a generic term for variety of closed-loop feedback servo systems, and "servo motor" as used basically exclusively within the RC model community. 

Note that "servo motor" does NOT specifically refer to the pulse-duration-controlled potentiometer-feedback non-continuous rotation "servo" actuators as used within the RC model and hobbyist community. It has broad usage in a variety of industrial control and CNC applications, the great majority of which would not be recognizable in any way to a person who thinks of a "servo motor" as the little things you put in a RC model or toy robot.



Anyways, fundamentally, a servo motor is the combination of a motor and a feedback mechanism, which is used together with a servo controller that controls power to the motor in order to control it's position. The controller, motor, and feedback system form 
a servo system.

Now, one thing you may realize here is that this is a very broad definition. This is true. In fact, if you add the feedback and control element to a stepper motor a stepper motor can be (part of) a servo motor! (I actually have a motion-controlled XY stage that uses steppers with optical encoders for feedback, and as such is "servo-controlled" for a project I'm working on).

The RC-servo mechanism that is so commonly conflated with the more generic servo-motor term is indeed a type of servo motor system, but it is a subset of term, not the entirety.

Probably 99% of industrial automation and computer-control uses drive mechanisms that fall under the "servo motor" banner, but they have much less exposure on the internet (it's a specialty field), so the hobby "servo" has come to dominate the common use of the term, and confuse the hell out of people who are just becoming interested in electronics.



With regard to your question, we need you to clarify if you are referring to hobby-style RC servos specifically or the more generic "servo" when asking your question. 

Realistically, a properly designed servo-motor system will outperform a stepper motor in every category aside from design ease, but your application may not need sufficient performance to make the additional effort worthwile, and a stepper system may be entirely capable of the task.

Furthermore, you can use a stepper motor as the motor element in a servo system, by adding closed-loop feedback around the motor (generally via an encoder of some sort).
However, steppers are normally used because they can often work well enough without closed-loop feedback, and that reduces the overall system cost by not requiring the additional encoders.

Once you have encoders, you can generally get better torque characteristics by using equivalently priced brushed DC servo-motors in place of stepper motors, with the control loop supplying the required precision that is lost by the use of the brushed servo-motors.