I believe in common usage, vivarium is basically a terrarium with live plants - does not have to have any sort of water feature, though it may. A paludarium has the live plants along with a water feature (pond, waterfall, stream, etc).
Per wikipedia:
"There are various forms of vivarium, including:
Aquarium, simulating a water habitat; for instance a river, lake or sea; but only the submerged area of these natural habitats. Plants in the water will use some nitrogen present within the system, and will provide areas for organisms to hide and forage.
Insectarium, containing insects and arachnids.
Formicarium, with species of ants.
Terrarium, simulating a dry habitat, for instance desert or savannah. A terrarium can also be formed to create a temperate woodland habitat, and even a jungle-like habitat. This can be created with pebbles, leaf litter and soil. By misting the terrarium, a natural water cycle occurs within the environment by condensation forming on the lid causing precipitation. Many kinds of plants are suitable for these habitats, including bromeliads, African Violets and Crassulaceae. Animals commonly held for observation include reptiles, amphibians, insects, spiders, scorpions and small birds.
Paludarium, simulating a rain forest or swamp environment. It also can be seen as an aquarium interconnected with a terrarium, having both the underwater area as well as the shore.
Riparium, a paludarium with circulating current through different-leveled pools"