You have the same fraud protection with a debit card as a credit card by same processor. However, the Fair Credit Billing Act limits your liability to $50 on a stolen credit card, while you could be liable for up to $500 on a stolen debit card if you don't report it within 2-days of discovering it missing. But many debit cards have the same $50 limit (and even that is frequently waived).
There may be more hassle with a debit card as refunds can take longer than credit cards, and it's your actual money tied up vs a credit balance.
Other down-sides of debit cards are that due to credit card rewards you're effectively paying an extra ~1-2% on all purchases if not using a rewards credit card and debit card usage doesn't help your credit score.
On the other hand, a debit card limits you to spending what you have, many people spend more with credit cards than they would have otherwise, which quickly negates the benefits. If you keep a good budget and aren't worried about falling into this trap, then credit cards offer clear advantages.
Reference: A Nolo article, Your Liability for Unauthorized Credit and Debit Card Charges