What is the ruling on benefiting from the exchange rate difference—the increase—when withdrawing money with an ATM card in a country other than my own, knowing that the card belongs to an Islamic bank?

Your withdrawal of funds using an ATM card from a bank other than your own in another country, with a currency different from your account's currency, is a loan from the bank owning the ATM, guaranteed by your account. There is no harm in this, as long as the loan does not involve usurious interest. Cash withdrawal from a source other than the card issuer is considered a loan between the cardholder and the withdrawing bank. The withdrawing bank then recoups the loan amount from the card issuer. There is no harm in benefiting from currency differences if no usurious interest arises from the withdrawn amount. As for your bank repaying the loan to another party in a currency different from the loan's currency, there is no harm in it, subject to its regulations.