There's a clear trend away from cash and toward credit card, debit card, and electronic peer-to-peer payments in the U.S. and in many other countries around the world. Investors have several different ways to play the so-called "war on cash," and two of my favorites are Square (NYSE:SQ) and American Express (NYSE:AXP).

While both of these companies are in the payment-processing space, they have very different business models and are also in completely different stages of maturity. Here's a look at both companies' businesses, their future growth catalysts, and which might be the better stock to buy right now.

Two young women paying for purchases with a credit card.

Image source: Getty Images.

About the businesses: How do Square and American Express make money?

Of course, there is more to both of these companies than I can discuss in a couple of paragraphs, but here's the short version.

Square's core business is developing payment-processing hardware systems for merchants, with a specific focus on small businesses. Traditional means of credit and debit card acceptance are prohibitively expensive for smaller merchants, and Square decided to capitalize on this opportunity. In addition to its hardware business, Square's business has evolved to include small business lending (Square Capital), a foodservice ordering platform (Caviar), and a peer-to-peer payments app (Square Cash) that is rapidly growing in popularity, just to name a few.

Square does not issue credit cards, nor does it serve as a lender to consumers -- not yet, anyway.

American Express, on the other hand, operates what's known as a closed-loop payment network. In addition to issuing credit card products and serving as the bank for its customers, American Express also serves as a payment processor in the same capacity as Visa or MasterCard. That is, American Express serves as the middleman between its cardholders and the merchants who accept AmEx credit cards.

The bottom line is that while these two businesses may seem similar on the surface -- both are engaged in payment processing -- they are actually very different businesses. Now, let's look at how each one is doing, and what could drive future growth.

Square: Impressive growth and some exciting new opportunities

In its relatively short time as a public company, Square has grown tremendously. Not only that, but growth is actually accelerating as the company gets bigger.

Over the past year, total net revenue has climbed by 48% and gross payment volume through the company's hardware solutions has increased by 30%. And if you think Square's growth could be headed for a slowdown, consider these points:

  • Square's gross payment volume represents about 2% of worldwide card payments. The company is in just five countries so far, and it is estimated that two-thirds of businesses around the world still don't accept card payments -- and most of them are small businesses. Plus, Square is having increasing success with large business customers as well, with standalone point-of-sale systems such as Square Register.
  • Square's Cash App customer spending on the Cash Card product has tripled since last December. This could be a massive driver of revenue growth over the coming years.
  • Square Capital originated 60,000 loans for a total of $390 million during the second quarter. While this represents 22% year-over-year growth, that means only 3% of Square's roughly 2 million sellers are taking advantage of the company's business lending so far. Plus, Square's long-term plan includes applying for a banking license of its own, so its future as a lender could be very bright.

As far as profitability goes, Square is definitely the loser in this comparison. Square has yet to turn a significant profit, and the company's 2018 guidance calls for a net loss. However, it's for a good reason. With such a massive long-term growth opportunity, Square is reinvesting the bulk of its profits back into the business and doesn't seem to have any plans to change.

American Express: A highly profitable business at a discount

American Express is by far the more mature of these two companies and is a very profitable business. With limited growth opportunities relative to Square, American Express doesn't need to reinvest everything it earns. For 2018, the company expects to earn a profit of about $7.10 per share.

Having said that, just because American Express isn't as exciting of a growth story as Square doesn't mean it has no room to grow. For one thing, the company is doing an excellent job of courting affluent clientele, something that's always been a strong point. Its high-end revamped Platinum card is resonating among the crucial millennial segment of the market, for instance.

Plus, American Express has a tremendous opportunity to expand its presence internationally. AmEx doesn't quite have the global reach of Visa or Mastercard yet, and in many places around the world, AmEx acceptance just isn't a thing yet. With an 18% year-over-year international consumer business growth rate, the company is doing a good job in this regard.

Which is the better buy?

It's difficult to pick just one of these stocks as a winner. I own both in my portfolio, and they aren't exactly an apples-to-apples comparison.

American Express is a tried-and-true winning business, and its P/E multiple of about 14 times this year's expected earnings seems extremely cheap given its solid growth rate. On the other hand, Square isn't profitable yet, but it's tough to find a stock with more long-term growth potential across several different areas of the financial-services business.

If I were going to add to one of my positions today, it would probably be Square, simply because I have a long investment time horizon and it's mind-blowing to think of how big Square's business could be in a few decades. Having said that, I don't think you can go wrong with either stock, and I certainly plan to hang on to both.

Matthew Frankel owns shares of American Express and Square. The Motley Fool owns shares of and recommends Mastercard and Square. The Motley Fool owns shares of Visa and has the following options: short September 2018 $80 calls on Square and long September 2018 $55 puts on Square. The Motley Fool has a disclosure policy.