First, what are the rumors? The Wall Street Journal has been out-in-front on both stories. On the Apple front, the paper describes the play as letting shoppers use their personal Apple mobile devices to make the purchases, with Apple leveraging "the hundreds of millions of credit cards on file through its iTunes stores."
The Apple move makes a lot of sense. First, Apple has experimented and tweaked this model over years while using it in its more than 400 Apple Stores. Other than Starbucks, that makes Apple the most experienced mobile payment retailer in the U.S..
The move has to first convince retailers to let Apple process these purchases for them. On the plus side for Apple, there are several reasons why retailers won't laugh the offer out of the room. First, Apple is a fellow physical retailer, which means that it understands the physical issues. Even better, Apple is such a laser-focused retailer (only its own products, plus a handful of very-related peripherals) that it is not meaningfully competitive with any other major chain. Apple has the experience to do this, both through iTunes (which would presumably be the engine used) and its own brick-and-mortars. The big issue is going to be the same issue that has undercut every other mobile payment plan thus far: interchange. If Apple can leverage its volume sufficiently to charge, let's say, Macy's meaningfully less than Macy's is paying the card brands and its current processor, it's pretty much a done deal.
That, however, is a truly tall order. That's partly because iTunes itself has to pay interchange on many of these transactions for all Apple customer who chooses to pay for iTunes purchases with a Visa, Mastercard, AmericanExpress, Discover or any other payment brand. Sure, some are tendered with debit, but it's hard to see how Apple can routinely charge a serious amount less than interchange.
That will mean it will fall upon Apple to convince the chains that the Apple clout will bring them more sales, which is also a hard argument to make.
Now let's look at Amazon. The Journal's report on Amazon said that the E-Commerce giant "plans to offer brick-and-mortar retailers a checkout system that uses Kindle tablets as soon as this summer" and that "in one scenario, the Seattle company would give merchants Kindle tablets and credit-card readers, the people said. Amazon also might offer retailers other services, such as website development and data analysis, the people said."
The Amazon plan really makes no sense if the target are the major chains. First, those chains detest Amazon (understandably) and absolutely see them as direct competition. Also, Amazon—more so than Apple—is obsessed with data-collection and the fear would be that Amazon would use the data collected inside Target and Walgreens to steal their customers. Even a powerfully low cost wouldn't likely make a difference. But both of these equations radically flip if the assumption is that the target are mom-and-pop stores. Although they are also Amazon competitors, it's such a different scale that it makes little difference. Amazon's clout could credibly bring in much more business to small stores.
More to the point, Amazon is offering its systems as a way to process payments and that's key for mon-and-pops. Apple's pitch is more about letting shoppers use their own equipment, which is much less compelling in a rural area where iPhones are rare, but Amazon purchases are common.