When you've booked valet parking, then all you have to do is drive to the departures hall. Here you will hand in your keys and car to a car park staff member, who will park your car for you. The main advantage of valet parking is the time you save. On your return to the airport you'll need to call the car park as soon as you've landed and a staff member will return your car to the arrivals hall.
With shuttle parking, you drive directly to the car park. A complimentary shuttle bus will take you to the airport. The benefit of this is that you can be certain that your car is safely parked. The shuttle bus will pick you up from the airport when you come back and drop you off at the car park.