On second thoughts you might have to bad keys. The keys have two separate parts, the security immobiliser and the remote locking
If one key has a broken/failed immobiliser chip, the car won't start, but the remote transmitter will still open the car.
The other key has a good immobiliser chip, which doesn't need any power, so it will start the car, but the remote transmitter has failed and so the remote won't work. Is the battery OK in that key? In the right way around? If you remove the battery in the key to check, you will have to remarry it by putting it into the ignition until the steering lock operates. So don't lock the car while doing the check.