First, they offer less protection than a credit card if the card is stolen or the number is lifted somehow.
Second, say you buy gas on a debit card and you only need $30 worth. Because it's debit they put a hold of $70 on your card. If you have less than $70 on your card you can get hit with an overdraft fee. But now that the federal gov't passed that overdraft protection law, gas stations can outright deny your debit card if you don't have enough to satisfy your hold. Hell, even if you did have enough money and needed to use that other $40 later that day, you can't because it's on hold.
You don't build credit with debit cards.
You might as well just carry cash with you, or use a credit card but don't spend more than you have in your account.