A regular servo is a type of gearmotor with some position feedback and electronics to maintain a set position.
There are two ways to modify a servo for continuous rotation. In the first method, you simply remove all the electronics, which leaves you with a pure gearmotor, as controllable (and as difficult to control; requiring power transistors of some sort) as any other gearmotor. In the second method, you remove only the feedback section of the electronics (actually, the connection between the mechanics and the feedback electronics), which leaves you with a motor that can be controlled as to on/off and direction with a simple digital signal (and to some extent, speed controllable as well, but this is somewhat ad-hoc and unpredictable.)
In either case, a servo is relatively common and inexpensive compared to an ordinary gearmotor (largely because the servo is a common "consumer" item, while the gearmotor tends to come from a company that doesn't want to sell to individuals.) If you build a project based on a "futaba XYX1234" servo, you can be relatively sure that other people will be able to duplicate the project (or that you'll be able to duplicate it yourself, a couple years later.) (This is in contrast with "surplus" gearmotors; the kind most likely to be available at a reasonable price to the average hobbyist.) Also, servos are available in a wide range of size and power ratings, ranging from 5g or smaller super-micro servos to 100g+ 1/4 scale servos with impressive torque.
The solarbotics.com motor offerings are sort of typical. Servo motors range in price form $6 to $13, while gearmotors range from about $7 ("toy") to $20 ("sealed metal")