Canon cameras have viewfinders that are blocked by the mirror in Live View – mirrorless cameras have electronic viewfinders that actually work while you record video. Older generation Canon cameras such as the 5D Mark III are limited to 1080/30p – mirrorless cameras record at 1080/60p (at least) for smooth slow motion. The Canon 5D Mark III lacks in-body stabiliation – the newest full frame Sony has in-body stabilization. All Canon DSLRs below the $7999 1D C are limited to 1920x1080p resolution. The newest full frame Sony records 3840x2160p resolution (4 times sharper) internally.
For just $499 more than the $2499 Canon 5D Mark III, you can get a modern Sony A7s II here: http://www.amazon.com/gp/product/B0158SRJVQ/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=B0158SRJVQ&linkCode=as2&tag=videomaker22forum-20
…and get incredible night handheld results like this:
https://vimeo.com/143513447 (shot handheld at 100fps)
Without in-body stabilization, the 5D III would need an expensive stabilizer to give you results close to this – and it would be completely unable to record high frame rate slow motion.
Sadly, the 5D Mark III was a great still/video camera in its day, but its day is past – at least in my view.
Hope this is helpful and good luck with your upgrade!
Hybrid Camera Revolution