The following is a script from "Fintech" which aired on May 1, 2016. Lesley Stahl is the correspondent. Shachar Bar-On, producer.

One sector of our economy after the next is being disrupted by new apps and websites, like bookstores, travel agents, taxis, hotels. Tonight, we're going to explore whether the banking industry is next on the list. Thousands of startups are challenging many aspects of banking, the newcomers argue that this important sector is too set in its ways. It's being called the financial technology -- or fintech -- revolution. We looked at the birth of one fintech company founded by two young fintechies who started not unlike the founders of Facebook and Microsoft.

Lesley Stahl: Which one of you dropped out of Harvard?

John Collison: That was me.

Lesley Stahl: And which one of you dropped out of MIT?

John Collison: By elimination--

Patrick Collison: Right, I was the other one.

Brothers Patrick and John Collison quit college because they had an idea for modernizing the financial industry they thought needed a shaking up.

Patrick and John Collison CBS News

Patrick Collison: In a world where people can send a Facebook message or sort of upload an Instagram photo and have it available to anyone anywhere in the world [snaps] like that, I think the fact that that doesn't work for money is something that seems kind of increasingly, honestly, unacceptable to people. And so, I think the question for banks is just-- can they get there first in providing these services? Or will it be somebody new?

They want to be the somebody new. John, 25, and Patrick, 27, first noticed the problem when they were in high school in Dromineer, a dot of a town in Ireland.

Lesley Stahl: And you were coders?

John Collison: Yeah, we had both learned to program growing up, and, we had been building iPhone apps, we had been building web services.

But when they wanted to charge people to buy the apps they hit an unexpected snag: they had to go to the bank and file paperwork just to be able to collect the money.

Patrick Collison: Like really sort of kind of like getting a mortgage. You'd have to, like, convince them that you were worth supporting--

John Collison: And like a mortgage, it would have to be approved.

Patrick Collison: Right, exactly. And it would take sort of weeks for this approval process to happen. And it just seemed sort of like this crazy mismatch.

So they decided to do something about it. They created software that allows businesses to cut through all that bureaucracy and instantly accept payments online from countries across the globe. We visited their startup, Stripe, in the Mission District, the heart of San Francisco's tech scene, where Patrick showed me how fast a business could set up a money-collection system using Stripe.

Pretend I left 60 Minutes to create an online business.

Patrick Collison: What do you want to sell?

Lesley Stahl: I think I'm gonna sell... dogfood. Homemade dogfood.

In five minutes, after a few clicks and a cut and paste of their code, he said my company would be ready to receive payment for homemade dog food online, right then and there.

Patrick Collison gives Lesley Stahl a demo CBS News

Patrick Collison: It doesn't need to take any longer. This is how it should work.

Lesley Stahl: And this is what would take weeks and weeks and weeks and forms and forms and verification and--

Patrick Collison: And going to the bank branch and waiting for paperwork to be mailed back to you and all this stuff.

They developed software for "buy" buttons, letting companies accept payments online fast and in new ways. Stripe charges sellers a small percentage for every transaction.

Lesley Stahl: Does the buyer pay anything?

Patrick Collison: The buyer pays nothing.

Lesley Stahl: Nothing?

Patrick Collison: Correct.

Their goal is to make money as easy to send as email - for everyone, anywhere, on any device.

Patrick Collison: We want to free businesses from just selling via credit cards, you know, to people who hold bank accounts, and instead, enable people to purchase online no matter what it is that they use, bank account or no.

"Many of the innovative services in financial technology that have come along in the past 10 years are not coming from banks."

Lesley Stahl: And of course this needed the smartphone, it needed this move to mobile.

Patrick Collison: For sure.

Stripe is hardly alone in inventing new financial technology -- or fintech. There's a revolution brewing with thousands of these companies trying to make banking faster and cheaper and increasingly mobile.

John Collison: Many of the innovative services in financial technology that have come along in the past 10 years are not coming from banks.

But by and large the newcomers are not challenging the core function of banks taking deposits. Even the startups themselves park the money they handle at FDIC-insured banks.

Patrick Collison: I think there'll always be a need for sort of somewhere to store your money, to have it sit. And, we think, you know, for all their flaws, they have a lot of experience at being banks, right?

But fintech is targeting nearly all the other functions of banking. The startups are peeling off one profitable service after another, typically offering them for less. It's called "unbundling the banks."

Say you need a loan: Fintech sites match borrowers and lenders directly the way Uber connects passengers with drivers. Need financial planning? Algorithms are replacing human advisers and brokers. Apps, like Venmo, let people click money to each other similar to texting and if you want to wire money across borders:

Taavet: I'm sending $500.

The CEO of a company called TransferWise showed us how his app can send money abroad and convert currencies -- say dollars into pounds -- without bank tellers and high exchange rates. Users just swap with each other.

Lesley Stahl: And a coupla clicks and boom.

Taavet: Click, click, done.

Lesley Stahl: Do you think that the big banks today see these fintech startups as the barbarians at the gate?

Vikram Pandit: Well, there's certainly a lotta curiosity.

Lesley Stahl: What about fear?

Vikram Pandit: There can be some fear.

Vikram Pandit, the former CEO of banking giant Citigroup, says it's the all-too-familiar tale of David and Goliath.

Vikram Pandit: A lot of what you're seeing in fintech is like what you're seeing with Uber or Airbnb. I mean, you've seen the impact of technology on travel or retailing--

Lesley Stahl: Yeah, the travel agents are-- were thrown out of business, that's what I saw. Is that what fintech is doing to banking?

Vikram Pandit: It's early days. And you know, banks are thinking about it, and they're trying to understand what all this new technology can mean.

It could mean trouble with millennials willing to ditch brand-name companies for new apps on their phone.

Max Levchin: The banks have not realized how different this generation is.

Max Levchin, who co-founded PayPal and was an early investor in Stripe, cites a survey saying 70 percent of young adults would rather go to the dentist than to a bank!

Max Levchin: They don't really have a problem putting their social security number into a web form, but they have a terrible problem going up to a teller in a bank, and trying to figure out what exactly you're supposed to do. "This is so inefficient. Why am I in this stogy outdated room that is empty and marble-laden?"

And it's not just about technology. There's also a question of trust.

Lesley Stahl: The millennials, their basically formative experience is the financial crisis. They're the ones who really don't trust the banks.

Vikram Pandit: And we know that many banks served their own interests more than those of their consumers.

Lesley Stahl: You're criticizing a system, basically, that you helped create.

Vikram Pandit: Well, there's no question the crisis demonstrated that the system didn't work. And when you looked at the aftermath of the crisis, what needed to be done. You had to make sure banks got back to the basics of banking, and that they had to address the trust issue.

But in the meantime fintech started taking root. In the last year and a half investors have poured over $20 billion into the sector including this banking insider who's personally invested in a dozen fintech startups. He says that beyond making banking more convenient, these companies can offer options to lower-income families that can't afford to bank at banks; 10 million American households don't even have a bank account.

Lesley Stahl: You know, I've read that it is more expensive for a poor person to use the banking system as it exists than for a wealthy person. How is that possible?

Vikram Pandit: There are bank account fees on your checking accounts. There are commissions, there are exchange rates. It all adds up.

Lesley Stahl: And that doesn't happen with the new companies?

Vikram Pandit: The new companies they're transparent and they tell you what the fees are. And they are fraction of some of the fees that are charged by banks.

John Collison: As services move onto the Internet they can provide the services more cheaply. And you know many of these banks, they have hundreds of thousands of employees. Whereas as we see financial services moving online, they don't have to have a physical presence and pay for that. So you can eliminate hidden fees if your cost structure is lower.

Lesley Stahl: And I'm hearing "eliminate jobs," I mean we're talking about hundreds of thousands of jobs in the banking sector, tellers and, you know, financial advisors, you name it.

Patrick Collison: I think in general technology always sort of makes some jobs less relevant, or perhaps, even obsolete, but I will say that the idea that sort of these people will find nothing else to do seems like it's way too pessimistic on the capabilities of everyone as human beings, right? These--

Lesley Stahl: Have you looked at the employment scene right now?

Patrick Collison: I think it'll take a while to adjust, but when you think about just the creativity of people and what they're capable of and the sort of aspirations and dreams that they have, the idea that they're not capable of anything more than sort of performing these automatable clerical tasks, I don't believe that for a second.

There are issues with fintech that go beyond the loss of banking jobs. Letting these new companies handle your money could be risky because there are concerns they're inadequately regulated. There's also the issue of online security.

Lesley Stahl: So why should we trust you? You think no hacker could ever get in there?

John Collison: I think we've had the best opportunity to sort of design a secure system, right?

Lesley Stahl: But you're still on the Internet.

Patrick Collison: Right.

Lesley Stahl: You are still operating on a base that's not secure.

Patrick Collison: I think you're right in the sense that people have been trying to steal money for as long as money has existed and, the best we can-- sort of-- as a society hope to do is to sort of design security in the most thoughtful and sort of robust way possible. And that's what we set out to do with Stripe.

And it's not like the big banks haven't been breached by hackers. So is fintech the next Uber? Well, it's still a small slice of the financial industry. And the powerful and rich old guard is fighting back, its lobby already pushing for more regulation to curb the newcomers; and - scrambling to adapt -- big banks have begun increasingly investing in and partnering with fintech, some looking at a technology called blockchain that's behind digital currencies like Bitcoin.

Patrick Collison: I think it's kind of human nature to always want to see these things as a competitive dynamic, that either technology companies have to win or the banks have to win and one of them is going to lose.

John Collison: It's not as black and white.

Patrick Collison: Yeah.

Lesley Stahl: Do you think what you have can be brought to a bank like Wells Fargo or JP Morgan, Chase? Can they integrate this or it's either one or the other?

Patrick Collison: I think they can be part of it, they can be part of sort of the infrastructure that powers it. And, again, we work with Wells Fargo and many other banks today. But I think that they can only be part of it. They can't be sort of the agents driving it forward.

He says that over one in four Americans online have used Stripe in the last year, including on sites like Facebook and Twitter, and department stores like Saks and Macy's. The software is embedded on both Apple Pay and Android Pay, and its already helped hundreds of thousands of businesses accept money online.

Even though Stripe has some stiff competition -- like PayPal -- the brothers have made two covers of Forbes and the 4-year-old company is now valued at $5 billion. Not bad for two brothers who not long ago had to beg their bank branch for approval.

John Collison: When you have a major technological shift like this, it's not clear that automatically, the existing financial players are the ones who are gonna win.

Lesley Stahl: Even though they're huge and powerful.

John Collison: I mean, there were plenty of huge retailers before Amazon, but somehow, this little, you know, upstart from Seattle-- you know, in just a few short years, gobbled up the business.

Lesley Stahl: Banks are so rich. Do you worry that they are going to come and buy you out?

Patrick Collison: Well, luckily we have a say in that. And we want to build a long-term independent company.

Lesley Stahl: Oh! You want to buy them out. (laughter)