In many ways, it's a battle of old versus new. For the former, we have incumbent credit card giant American Express (AXP 2.27%), a company that is still based on the traditional swipe-to-pay credit card transaction, complete with a fixed credit card station.

IMAGE SOURCE: TAXREBATE.ORG.UK, VIA FLICKR.

In the opposite corner is Square (SQ 4.50%), a 21st-century creation with a core business that also consists of processing credit card buys. The modern twist is that Square uses a proprietary reader that plugs into a smartphone or other mobile device.

IMAGE SOURCE: SQUARE.

American Express has been a mainstay among finance industry investors for decades -- it's a longtime core holding of Warren Buffett's Berkshire Hathaway, for example. By contrast, Square is a newcomer, having hit the market in an IPO late last year.

Both companies would very much like to be your credit card processor of choice. As far as investors are concerned, though, which of the two is more deserving of a buy?

AmExclusive no more
Neither stock is an investor darling just now. AmEx is still reeling from being replaced last year as the exclusive credit card brand accepted at popular retailer Costco Wholesale, not to mention its departure as the co-branding partner for Costco cards.

That was a painful blow, as the retailer's business comprised around 20% of AmEx's total loan book. Compounding that, airline JetBlue also ditched AmEx as its co-branded card partner around the same time.

The company is trying to make up for those losses by signing up new partners. It's had some success, inking a co-branding deal with big-time brokerage Charles Schwab and winning acceptance for its cards to be used for purchases at Wal-Mart's warehouse outlet chain Sam's Club.

But you don't get over the sudden elimination of 20% of your loan book overnight. With the looming demise of the Costco deal (which officially expires next month), AmEx suffered declines in both net revenue and bottom line in its fiscal 2015. The former totaled $33 billion, 4% below 2014's result, while the latter fell 12% to $5.2 billion.

The immediate future doesn't look much better; stripping out the gain from the pending sale of the legacy Costco co-brand portfolio to Citigroup, AmEx anticipates profitability will again decline in fiscal 2016.

Squarely in the red
Square hit the market in one of the flashier IPOs of 2015, but its stock hasn't burned brightly since. It's down by about 10% from where it closed at the end of its first trading day.

Like many young tech companies, it's posted a sharply rising top line, but deepening net losses. That's because costs have been increasing faster than revenue; they were almost 350% higher in fiscal 2014 (the latest fiscal year reported by the company) than in 2012, while top line grew by 318% across that stretch of time.

When it first arrived on the scene, Square seemed fresh and revolutionary -- finally, small merchants and street corner vendors could easily accept card payments.

But technology moves quickly, and these days Square is far from the only solution in town for such businesses. PayPal Holdings, for one, has a reader of its own, and private companies like North American Bancard provide similar goods. More competition is sure to follow; this is not exactly a high barrier to competition.

Square has labored mightily to get its hooks in complimentary segments like big retailing -- securing a card-processing deal with Starbucks, most notably -- and small business lending.

But the Starbucks deal has been a consistent money-loser, and those other lines face plenty of competition from a slew of better-positioned, more experienced rivals.

Ringing up a winner
For me, neither of these stocks offers outstanding value just now. But of the two, AmEx has a stronger core business by far and better prospects for long-term growth. The company will get over the recent business losses eventually, and on the way it should continue to post a profit as it ever does.

We should also keep in mind that AmEx pays a dividend. At a yield of 2.1% on the current stock price it's nothing to dance and sing about, but it at least puts some money into the pockets of shareholders. The same can't be said for Square, which isn't in financial shape to make such a payout, and very possibly never will be.

So if forced to choose between the two, I would pick American Express -- with reservations, however, as I think the company will be in recovery mode for some time.