Debit cards differ from credit cards in one major way: the amount is automatically withdrawn from your bank account. While credit cards give you several weeks until payment is due, debit cards reduce your account at the time of the transaction (depending on the merchant and the timing of the purchase).
Another difference: you never (or rarely) receive rewards for using debit cards.
One positive aspect to debit card use over credit card is that it's more difficult to get yourself into enormous debt. This is because you withdraw against existing balances in your account, either savings account or checking account (check with your bank for details)...if the money doesn't exist, you can't withdraw. (This is unlike credit cards which extends a line of credit to you allowing you to use money you might not have).
If you are budgeting your money and trying to eliminate debt, using a debit card can help. As long as you don't use more than you have (which might lead to fines against you) it's a great way to learn to budget. Credit cards, on the other hand, allows you to spend money that you might not have!
If you are a shop-a-holic, consider destroying your credit cards and using debit cards only. That way, you only spend what you have available in your banking account. Better yet, pass on that purchase!