I had never heard of a "vivarium" until I started posting here, but it makes sense to me...
IMO, tank is a generic term for the container... the [fill-in-the-blank]+arium tells you what is inside the tank.
Aqua= water, Terra = earth, and Viv = life
...so to me, a viv would suggest a living creature, as opposed to just plants. (of course, I know plants are living things too, but I don't need to split hairs. lol)