LINUX, UNIX…what’s the difference ?

I’ve recently been telneting to a public access unix system (PAUS) to learn UNIX, my purpose is to learn about the working enviornment of UNIX and shell progming, NOT admin. At least not yet. But the host is actually running LINUX….there isn’t onesingle reference to this fact on their website, they just call their system UNIX…so my question is: should I care ?
Does this make difference to what I’m pursuing ?
Thanks