Payment-processing company Square (NYSE: SQ) has shown tremendous growth over the past year or so, and its stock price has roughly tripled as a result. And credit card giant American Express (NYSE: AXP) has rebounded nicely after losing its Costco partnership and is growing strong once again. Let's look at the business models of these two companies and see which might be the better addition to your portfolio now. How Square makes its money Square's core business is making payment processing systems for small businesses, and earning a profit from transaction fees. When a merchant takes a card payment using Square's hardware, Square gets a percentage of the transaction. From this "swipe fee," some of the proceeds go to the card network (Visa, Mastercard, American Express, or Discover Financial Services), while some goes to the card's issuing bank, and Square gets the rest. Image Source: Square. In addition to its core business, Square also has several other businesses, such as its Square Capital small-business lending platform, which is showing tremendous growth already, and its Square Cash peer-to-peer payments platform. Square aims to leverage its payment-processing solutions into an ecosystem for small businesses' financial needs. How American Express makes its money American Express is a "closed-loop" payment network. Unlike Visa and Mastercard, which don't lend any money to consumers, American Express serves as both the card network and the issuing bank. Therefore, the company makes money both on swipe fees and interest charged on its outstanding loans. AmEx also makes money on card fees, such as the annual membership fees that its higher-end credit card products charge. Growth potential Of the two, American Express is by far the more mature of the two companies and therefore has somewhat limited growth potential. There is the potential to add new and innovative credit card products, create more co-branding relationships, expand into more international markets. And of course, as the economy grows, American Express' revenue can grow along with it. On the other hand, Square is a leader in a largely untapped market. Two-thirds of businesses worldwide still don't accept card payments, and Square is only in five countries thus far. Worldwide card payment volume is expected to reach as high as $55 trillion annually by 2025, so it's fair to say that this is a big growth opportunity. And don't forget about Square's other operations. Square Capital alone has the potential to be a multibillion-dollar revenue stream for the company. To sum up the growth potential of these two, American Express is a company that can reasonably be expected to grow its business at about 10% per year, and as we'll see in the next section, this is about what analysts are expecting. On the other hand, Square is a rapidly growing company whose revenue could easily double or more within the next few years. Valuation As you might expect, American Express is the "cheaper" of the two in terms of valuation. The company trades for 18 times trailing-12-month earnings, which is quite reasonable considering that analysts are expecting 10.6% earnings growth in 2018 and double-digit growth over the next five years. Square has just recently become a profitable company, so looking at traditional valuation metrics such as trailing-12-month P/E aren't meaningful. And the company trades for 82 times its expected 2018 earnings. In a nutshell, American Express is a value stock while Square is a growth stock in every sense of the term, so a valuation comparison isn't terribly useful. Which is the better buy now? It's tough to name a clear winner here, simply because these are two very different businesses in different stages of maturity. And in full disclosure, I own both in my personal stock portfolio and I don't think investors can go wrong with either company. In addition, these stocks fulfill different investment objectives. American Express is a rock-solid company with predictable earnings, revenue, and cash flow and is certainly the safer bet. On the other hand, Square is a much more speculative investment and isn't suitable for investors without significant risk tolerance and the stomach to deal with the ups and downs along the way. Having said all that, if I were to buy shares of one of these companies today, it would have to be Square, because of its incredible long-term growth potential and its ability to create a commerce ecosystem for small businesses. 10 stocks we like better than American ExpressWhen investing geniuses David and Tom Gardner have a stock tip, it can pay to listen. After all, the newsletter they have run for over a decade, Motley Fool Stock Advisor, has tripled the market.* David and Tom just revealed what they believe are the 10 best stocks for investors to buy right now... and American Express wasn't one of them! That's right -- they think these 10 stocks are even better buys. Click here to learn about these picks! *Stock Advisor returns as of October 9, 2017Matthew Frankel owns shares of American Express and Square. The Motley Fool owns shares of and recommends Mastercard and Visa. The Motley Fool owns shares of Square. The Motley Fool recommends American Express. The Motley Fool has a disclosure policy.