In all of the semi-manual modes (aperture-priority, shutter-priority and program auto), you set one or more settings manually. The camera then chooses the rest of the parameters automatically to give you a nominally correct exposure.

However, sometimes you want to override the camera's metering, either because it wouldn't correctly expose your subject or because you simply wish to take a creative decision to expose differently. In these kinds of cases, without exposure compensation, you would have to change to full-manual mode if you wanted to retain control over exposure.

With exposure compensation, the camera under- or over-exposes the image by the number of stops you dial in. However, the aperture or shutter priority remains in effect, allowing you to creatively control the exposure without losing the convenience of the camera selecting some of the settings for you.

This is both convenient and can save quite a lot of time, allowing you to get shots that you might have otherwise missed.

TL/DR; Use exposure compensation to adjust exposure in the semi-manual/creative modes. Use full manual mode when you need total control over all the settings (aperture, shutter and ISO).