I never use a debit card, especially for automatic transactions.
If there is a screw up with a debit card (say they charge you $1,370.00 instead of $137.00) they have YOUR money, and it's your problem to get it back. Naturally they won't pay you interest or the bounced check charges you incurred due to their error.
If you get in a dispute with them, they already have your money (and can instantly take even more of your money if they want to). You then have to a) Settle the dispute in your favor and b) get your money back. In this sort of situation you have far more negotiating leverage if they do not already have your money.
Credit cards allow you to dispute charges through the CC company...you don't have to pay until it is resolved, and the CC provider will back-charge the other party to reimburse you if necessary.
If you pay it off every month, and have a CC with some kind of rewards program, you'll be ahead of the game.