I have licenses for 2 versions of cubase elements 8 came with my MOXF8 and AI 8 came with a Steinberg audio interface. Are these versions of Cubas esentialy the same? If not which one would be best to run in a home studio?

I’d suggest you go with Elements, since AI is simply an introductory version. You can compare the commercial versions of Cubase here:
https://www.steinberg.net/en/products/cubase/line_up.html

Cubase Elements and Cubase AI are almost exactly the same. There is a third version called Cubase LE which comes bundled with some non-Yamaha hardware.

In fact the installer since at least v7 is actually for all three versions !!?? which one you actually get depends on the license.

All three are “light” versions missing many of the features of the full version (now called Cubase Pro).

Cubase Artist is an intermediate version, more features than Elements/LE/AI but less than Pro. Why anyone would bother to buy it is beyond me.

Cubase LE is not a commercial version of Cubase and support for it is now limited. The link I provided shows the differences between the commercial products.

Thanks, I appreciate the replays. Exactly the info I needed.