Basically, the Durbin Amendment caps interchange on debit cards at a far, far lower rate than credit cards; they also have different networks (not MC or Visa) that can be run at least for in person transactions. However, the limit does not apply to banks with less than $10 billion in assets.
This is why some online banks can offer lower fees and sometimes even debit cash back (1% is the most I've seen though, so not competitive with credit). But its also why ever fintech under the sun pushes debit cards so heavily--the debit card is issued by a (small, non Durbin-covered) partner bank who then shares revenue with the fintech company.
At the end of the day, credit cards are still significant better for customers. You can have multiple cards with different benefits while keeping all your money in one place, and you can prevent overdrafts because you don't have random subscriptions taking out unpredictable amounts of money in the middle of the night and causing a check to bounce the next day. You are no longer obsessing over "early payday" features or ACH speeds.
Credit cards also provide a nice hack to allow cash deposit for those with only online banks--you can often use a big bank's ATM to make a payment towards the credit card, allowing you to get rid of ordinary amounts of cash.