In several ways
1)Debit cards don't build credit history. This makes it hard to get a car or house loan at good rates.
2)Credit cards have 0% interest if you pay at the end of the month every month.
3)Debit cards do not earn you interest. If you have an interest checking account (rare, and usually such a low rate that its a joke, sub 1% in most cases), you earn that money regardless of if you have or use a debit card.
4)In the US, many purchases such as hotel, rental car, and gas put a hold on your account for more money than the actual charge. This hold goes away once the car is returned/hotel is checked out/a few days (for gas), but in the meantime that's additional money you can't access.
5)Emergencies/hard times. Sometimes shit happens. You may lose your job and run low on cash. You may have a series of car and house repairs. Its always a good idea to have an additional emergency fun you can call on for short term cash.
6)Your bank may put a hold on your debit card for suspicious activities. In that case, your card is useless. Having a backup is always a good idea. There's been several times this has saved my ass when traveling.