He's giving tae min a choice and time to open his eyes and accept the fact that he died and now is in another body. Tae mins personality is the kind that when he has no exit he will forcefully make one and run away so Si woon said that if he stays and never tells the truth (that he is tae min) he can get away therefore a way out so hes less likely to try to forcefully get away from Si woon and also it gives Si woon time to try to convince Tae min to accept that he's dead and to trust Si woon once more and tell the truth. And more time for Tae min to think things through and to slowly accept his new life and it would probably be an issue if people knew that he wasn't Han Si Won (newspaper laboratories yada) worse case scenario. And tae min really doesn't trust Si woon, Tae min thinks Si woon abandoned him when they were children and that he's playing with him lying saying he cares. Tae min although doesn't look like it he's going to fall apart eventually. He's desperately trying to protect himself from getting hurt by Si woon again and keep people from finding out that he's not si won also was probably holding out hope that he could get to his old body again and was trying to keep things on the down low to keep suspicion off himself by saying he was Han Si won just with amnesia.
Wait... I didn't get what White Hair was saying at the end. If WH already knows that Black Hair is actually Brown Hair, then why does Black Hair have to hide it? What's with the bet? Is he talking about hiding it from the rest of the world? I doubt it would be really an issue.