The more obvious difference is that adjusting any settings does not do any compensation to the camera's exposure meter.
In Aperture priority if you stop down the aperture, then the shutter would decrease to bring the image back up.
In Shutter priority if you speed up the shutter, the aperture would open up to bring the image back up.
In Manual if you stop down the aperture the camera's exposure meter would tell you that you're under exposing.
Dialling in exposure compensation biases this selection to change the ultimate brightness of the exposure meter in the camera. It will then purposely pick an aperture or shutter speed that darkens or brightens the image, and in manual mode it will tell you you're correctly exposing when you dial in the correctly adjusted settings.