Type I error is also known as a "false positive" and type II error as "false negative."
Example of Type I being more serious: Suppose you have a banking app that gives access to an authenticated user to transactional capability. In order to authenticate the user, a biometric test is done. If the biometric test makes a Type I error and falsely tests the user as positive (correct user), then an imposter who just got in, can transfer money away from the account. Otoh, if a true user gets rejected (type II error), it is a nuisance but not as serious as the loss of money.
Example of Type II being more serious: Suppose you have an app where a doctors can access patients' blood types. In an emergency, a doctor that fails to gain access due to a Type II error (false negative) won't be able to help the patient until they get an additional test done, which can be a serious delay. In this case a false negative is more costly than the problem of letting in an impostor (false positive).