Hello,
In the world of Linux, there are multiple Linux distributions to choose from. Some of these come from commercial vendors, some are entirely free and community-supported, and others are a hybrid of the two.
For a bank, I'd imagine you'd be looking at a fully-commercially-supported offering. The closest thing there is to a default choice in the commercial Linux world would be Red Hat Enterprise Linux (RHEL). This is a version of Linux provided by Red Hat, who are in turn owned by IBM these days. You pay for support, for access to updates, and do so on a ongoing basis as part of a subscription.
The next-most-common commercial Linux distribution is SUSE Linux Enterprise Server (SLES). This is provided by SUSE, a German-based Linux company which, like Red Hat, has been going since the 1990s. In Continental Europe this one has a very big following, and was one of the first successful commercial Linuxes there. Again, they provide support and updates on an ongoing basis for a recurring fee.
Another commercial option worth mentioning these days is possibly Ubuntu, from Canonical. Now normally most people tend to run this one unsupported and as-is, since it's one of the (many) Linux distributions that are available free of charge and which you can run and get regular ongoing updates for at no cost. But Canonical do offer commercial support as well. I've no experience myself with their paid-for offerings, but you could look into this.
One last one that's probably worth mentioning is Oracle Linux. This one is available for free to download, install and get updates for - you only have to pay Oracle if you actually want a support contract. This began its life as a re-spin of Red Hat, and still basically is that under the hood - it's really just Oracle's take on RHEL, more or less. So if you have an existing commercial relationship with Oracle, and don't want to set up a new contract with Red Hat/IBM, looking at this could be worth your time.
Outside of commercial offerings, your choices are near-limitless. Debian, Ubuntu, CentOS, Fedora..the list goes on and on. But given that you've mentioned that you're working in the financial industry, you might well find that one of the commercially-supported offerings mentioned above is the best way to go (or at least the way that'll be approved for use).
In closing, you ask what the difference is between Linux and UNIX. UNIX is a trademark that indicates that a given operating system descends in some fashion from the original AT&T UNIX from back in the day. Linux, interestingly, cannot claim such a direct line of ancestry - strictly speaking, Linux is not a UNIX operating system, but is very much UNIX-like, and is effectively a UNIX in all but name (and in all but legal fact). A full discussion of why and how this all came about would take longer than we have here, but suffice to say one of the design goals of most Linux distributions is to be as UNIX-like as possible, without actually being UNIX.
Anyway, hope this helps ! If you have any further questions let us know and we'll see what we can do.