Suppose that there is Mr. A, who received a plastic surgery to have the same face as Mr. B and changed his name to Mr. B. He uses this name for years and everyone knows him as Mr. B now. I then get to know his current face and name and write that in the Death Note. Who will die?
My personal guess would be that nobody would die. It was established at this question that despite changing names, the Death God would still see the original name. Therefore Mr. B would still show A as his real name. So if you would write down Mr. B, while thinking of Mr. A, the names would not match.
Now about having the exact same face. One could argue that Mr. B would die, because you did write down his name and did think of his face, but despite having undergone plastic surgery, your face will never be 100% exactly the same. Even with identical twins, they will have differences. That's why I believe that nothing will happen, since the Death Note probably knows you are not thinking of Mr. B, but of Mr. A, due to their faces being slightly different.