I've done a number of these flights. There aren't any catches. If you've got friends who've done AFs also, you probably already know the basics, but a couple of items: (1) go/no go is always up to you. Passengers are told that these are GA flights, and that weather and other factors could present last minute problems. They are encouraged to make backup plans just in case. (2) Angel Flight America is the broad umbrella name that encompasses a number of regions. They are all independent, but work together. The independent part means that each region is free to set its own rules, so the requirements to be an AF pilot can vary. Some require only the 200 hours mins specified by the regs, some require 300. Some require an instrument rating, some don't. Just be sure to check on the requirements of the region for whom you'd be flying. (3) It's far better to own your plane than to try to do these with rentals. Not that it can't be done, but rentals are more likely to be down and coordinating the flight and the rental can be a problem. (4) There are volunteer liability forms that passengers sign prior to boarding the plane. It's important to bring a stamped envelope and leave these with the FBO for mailing BEFORE you begin your flight. It basically is their acknowledgement that this is a volunteer flight and releases you from any liability should anything happen. (I've always wondered how these would stand up in court, but that's another thread!)
As for deductions, they are considered charitable donations and are as Tarp already indicated, although I'm not certain how the aircraft rental works. As mentioned, if you own your aircraft, you can't deduct pro-rata expenses for things like monthly hanger fees, loan payments, normal insurance, etc. How much the deductions are worth to you personally will depend on your income. If it's high enough, zero will be deductible.