Analog servos have a position feedback system in them too. They have to or else they would have no corrective ability if you try to deflect them away from where they have been directed to go.
Digital servos are constantly powering the motor to hold a position, whereas analog will only power the motor to make an adjustment to position (once they are at position they will stop powering the motor). This does result in much quicker transit times and latency specs since the motor is designed to have very low inertia and is more or less always under power and moving in one direction or the other. Size for size, the digital servos also have better motors in them (coreless/brushless) and much narrower "deadbands" (deadbands being where minute changes in the position are not detected and thus will not trigger a correction). Digital servos also offer a wider, faster range of input signalling which helps to achieve that faster, more precise control. A result of all this is that digital servos chew up more power and thus need better cooling strategies in their design, but for the precision and speed you get with it, it is deemed worth it.
Why are they more expensive? More complicated electronics (micro processor control as opposed to analog or timer based control), better motors and position sensing hardware, more design time and usually better materials in order to deal with the heat build up and to keep motor core and gear train inertia in check, better transit times and latency specs, higher frequency input handling (usually), etc.