Short answer - tokenization and eCommerce security.
Chip and Pin uses physical contact of the card to the payment terminal. The chip is very hard to duplicate so it essentially eliminates card cloning. The PIN provides a second factor to authenticate a trusted customer at the point of sale.
Apple Pay is a variant of NFC (near field communication) much like Google Wallet with PayPass. This is wireless (contactless).
The specifications for hardware (Level 1) and software (Level 2) for both contact and contact-less payment systems are managed by EMVCo; see
http://www.emvco.com/. Apple has chosen wisely to work with existing industry standards with a couple of big improvements. These guys are smart and they got it right.
What is different about Apple Pay is that unlike PayPass and Chip&PIN which send the credit card account number through all links in the system (usually encrypted), a one time use token is created. If an Apple Pay transaction is exposed, only a useless one time token would be divulged. This is much like the DUKPT system used from other secure transactions. If you are curious about DUKPT see en.wikipedia.org/wiki/Derived_unique_key_per_transaction.
Aside from the technical superiority of tokenization for security over Chip&Pin for card present transactions, how Apple Pay is better is a matter of taste because both systems mitigate card present fraud. It will always be hard to argue that either will be faster than a swipe.
I'm sick of sitting on my fat wallet and if I could get rid of my cards that would be great. Unfortunately I'll still need my cards because most places wont take anything but mag stripe for years (its taken Canada 7 years to get to 90% Chip&PIN at the merchants). If I lose my wallet I have to call a bunch of companies, whereas if I lose my phone nobody is going to get my card numbers. Apps will proliferate that allow loyalty programs and discounts to be offered to me through mobile integration (e.g. Amex has already announced buying McDonalds food with points). Geofencing in conjunction with NFC will further promote targeted marketing and I like the idea of saving money in exchange for loyalty. NFC will also likely be used to do things like open doors, start cars so it will be very convenient.
Apple Pay also supports tokenization for eCommerce. This is huge because card not present is the area where credit card fraud is most persistent. Apple Pay and the variations that are sure to follow will knock the online card fraud guys out of the game.