I should declare from the start that obviously I think everyone should do both - they totally compliment each other - but say you only have time in your life for one. Which one is right for you? Yoga or Pilates?
Pilates is a system specifically created to strengthen the body and to ensure you retain the functionality of the joints (especially the spine) for the rest of your life. (For as long as you practice Pilates regularly)
Yoga is a system that works your body but with the outcome of helping you to practice relaxation and focus. You make shapes with your body (poses) that help you to unite breath, mind and body using your physicality as a laboratory.
Both, yoga and Pilates use breath and require focus which inherently brings relaxation of the mind but for one that is a bi-product (Pilates) whereas for the other, it is the whole point (yoga).
Pilates is more concerned with functional movement and can be used in a rehabilitative way with modifications. Yoga is much more of a bit of contortionist fun and though, in this day and age, there is alignment and safety, it is more about simply moving your body.
Both systems have their merits and compliment each other, so try both before making your decision (if there really is only time for one).