Is the husband obliged to return the second month's rent money to his wife's mother, which the landlord did not demand, knowing that the money was given to him to pay the rent?

Your husband is not considered a thief. He must pay the agreed-upon rent to the landlord. If he rented for less than what your mother paid, he must return the excess amount to her, unless she waives it for him. This is because he is considered an agent, and any excess obtained by an agent belongs to the principal.