I read the puzzle of identical twins and I don't think I understand all parts of it.

There are 2 identical twins, one always tells the truth and the other one who always lies. The truth teller is accurate in all his belief, but the lying brother is totally inaccurate in his beliefs.

When we ask both of them the same question, they both give the same answer, which means they both are telling the truth. It's just that one of them has mixed up mind and thinks that he lies about everything, but in real he is also telling the truth.
Now, calling him a liar wouldn't be true, because even though he thinks wrong and says the opposite, he's still telling the truth.
But, when we ask him if he is the accurate truth teller, and he says "no"- wouldn't really be true, because he is telling the truth to the other questions. The answer "no" would be true just for the people who think that that's the answer from a person they think has problems with his head.

If they both are telling the truth, then why would you want to find the liar??

I would appreciate if someone tell me if I'm thinking right.
Thanks.