Personal injury liability provides coverage for claims asserted against a negligent driver. So, if you are injured by a driver who was drunk, speeding, or texting behind the wheel, you would file a claim against his or her policy for personal injury liability (you would likely also file a claim for property damage liability as well).
On the other hand, medical payment coverage provides compensation to the policyholder and his or her passengers. So, along with seeking personal injury coverage under the negligent driver’s policy, you can also seek med pay coverage under your own insurance. Note, however, that if your personal injury claim is successful, you may be required to repay your med pay benefits.