The closest parallel to bank linked debit cards is a current account cheque.  Like debit cards, cheques draw funds straight out of a current account and they don’t have a separate account.  There are some differences.

Debit cards offer quicker confirmation that the funds are in the account.  This means that there is not the phenomenon of bounced cheques, when a shop keeper can find that they have no money even though the goods have left the shop, making recovery considerably harder.  This has meant that many shops have stopped accepting cheques due to the frequency in which they are not honoured.

Debit cards also offer a faster transfer of funds.  This does not just affect the merchant, but also the card holder.  This means that the old practice of writing a cheque before pay day in the knowledge that it would take some time to arrive and to be cashed is no longer possible.  However the consumer has a clearer idea of how much money that they have in their account and are less likely to be caught by too many payments being made at one time.

Where the cost falls is also different.  When a consumer writes a cheque then the shop keeper will not be charged.  In these cases the consumer would be charged.  In some cases the consumer may get the cheque for free, but in the end the consumer would be liable for it.  With a debit card there is a merchant charge that the shop keeper or service provider is charged.  There may also be a transaction charge that the consumer would have to pay but essentially the charge falls mainly on the person who sold the good or service.  This can reflect in slightly higher prices, or discounts not being available for cards.

Another difference is that more people can take cheques, as essentially all the person needs is a bank account, whereas a debit card needs a card reader or an internet merchant account.  The amount of people who accept debit cards is essentially limited, which means that many smaller businesses such as taxi drivers or market stalls will not take debit cards.

Debit cards are also harder to commit fraud with as transactions need to be confirmed with the bank before being processed and banks tend to have better ideas of when a credit card is being used in unexpected ways (for example being used abroad).  However a stolen debit card cannot have the transactions cancelled as easily as a cheque can be.