The new style UK driving licence consists of two parts - a photo card and a paper licence. The licence is not complete without BOTH parts. You are advised to take both parts when collecting your vehicle as you may be asked for them both.
Often the rental company will only ask to see the photo part, so in most cases you will be OK to drive, but in the event they do ask for the paper part (which lists any points and endorsements) and you don't have it with you, then you will not be allowed to drive, nor will you then be entitled to a refund on your rental.
If you do not have the paper part and the rental company do not ask for it, so you collect your car, then you also run the risk of problems in the event you are pulled over by Highway Patrol and asked to show your license, as they may insist on seeing both parts. As it is a legal requirement to carry your FULL license with you in the car at all times when driving, then you could face legal consequences