The difference isn't the OS, it would be the specific shell, be it bash, korn, borne, or C shell.

Which Unix should you use? Well, that depends on what matters to you. There are people with very strong inherent prejudices, and there are also occasionally good reasons to prefer one flavor of linux or unix over some other.
For the most part the noticeable differences are minor and annoying and not ones that will plague you in learning shell scripting. As secondarypencil says, it's more about the shell than the OS. The differences between OS's tend to be in things like where certain system files live and stuff like whether they use upstart or init or whatever to get everything up and running. Actual day-to-day operations will pretty much work the same on all systems, and where they differ, you just learn how the next thing works, so it's not really a big deal.

So go ahead and use whatever you like. Go with Ubuntu if you can't think of any reason to use anything else, if only because it's the most popular distro and you'll be familiar with the machines your linux-using friends bring to you.

The important question is, what's your toolkit? bash is always a good choice, it's on everything. perl is a good one to have in your bag, if you want to be beardy and mysterious. If you want serious juju points, learn awk. People won't fuck with you if you can do awk magick - they'll figure you've got the Black Arts, and you can call up demons from the vasty deep or something. It's pretty useless if you know perl, but it still impresses the shit out of people.

When it comes to Linux vs Unix scripting, there can be no difference depending on the *nix you go with. In general, Linux has more GNU. However, most of the open source BSD will have that as well. The only time you'll be walled in is the older proprietary stuff, Solaris, HP-UX, etc.

I'd take the Linux you're comfortable in and just use that. For extra realism, make sure /bin/sh actually maps to Bourne and not bash.