concede, admit

[ShadeWe]

‘That’s the only possible solution.’ ‘Yes, I suppose so,’ Charles conceded/admitted

conceded has been written here on default.

What's the difference between these two?
What meaning does each word has? I've tried to look up in a dictionary, but the words' explanations are confusing to me. They are very similar and I can't completely get the difference.

[bhaisahab]

In that context there is no difference of meaning. Say: 'What difference does each word have?'.

[ShadeWe]

What difference they have beyond this example? I guess concede carries the meaning of giving up, or something. For example: 2 persons are arguing and either of them says: all right dude, I concede you are right.

Is it correct to think this way?

[emsr2d2]

What difference do they have beyond this example? I guess concede carries the meaning of giving up, or something. For example, 2 persons people are arguing and either one of them says "All right dude, I concede. You are right."

Is it correct to think this way?

Yes, in the context you gave, "concede" would suggest that one party has given up trying to convince the other.

[andrewg927]

Admit has other meanings like you got admitted to a college. You can't use concede there.