Which is better to use : Mirror or totally reflecting prism?

Based on an update in the comments it looks like you're trying to build some kind of stereoscopic mount or device to capture two images in one exposure.

A prism offers better light transmission. Mirrors are significantly lighter. Mirrors offer some ability to modify their geometry. Mirrors are significantly cheaper to produce.

Which is right for you depends on the circumstances you wish to satisfy.

All the prior art I found (albeit in a very very brief search) in the 3d photography field were mirror based devices such as the Kúla Deeper. I have to assume that the significant weight and cost differences won out over the better optics.

Prisms have fewer air-to-glass transitions than a series of mirrors, and therefore better image quality. This is why SLRs and DSLRs with pentaprism viewfinders are usually preferred over pentamirror finders — although the latter are lighter and cheaper, both of which can also be significant advantages.

First thing to do is to walk up to a mirror and look at your reflection. What do you see? Do you see one nice crisp image? No you dont. You see 2 images. So basically you need to find a surface silvered mirror like the one in a SLR camera or use a prism and deal with refraction . Unless you know optics the former will be easier but alignment is critical.