Anthonyfear did a great job comparing the main differences between the two tablets. I would say the main, and most important difference, is the processors and what it limits the device to. The ARM based Vivo RT, is limited to programs and apps from the Microsoft App store. You will not be able to download anything from a software CD, USB drive or eve a website. Think of it like a closed-off ecosystem, like the iPad.
The Smart, on the other hand, uses the ATOM processor. The ATOM processor is a little faster, but because it uses this, the tablet is able to install and run legacy software and full Windows 8. You can download and install anything that you would be able to on a regular computer/laptop running Windows 8. It does have an available keyboard, although it is not as nice as the VivoTab RT's hard-connected keyboard that extends the battery life.
There is regular VivoTab (TF810C), that is sort of a combination between the Smart and the RT. It is a little bigger than the RT, and runs full Windows 8 from an Atom processor. Unlike the Smart, there is an available keyboard/docking station available for it.