What would an MRI “see” that a CT Scan would not and why?
Depends. The ordinary version may, perhaps, give you better contrast with certain tissues. The contrast with X-rays depends on the density of nuclei, and the heavier the better, which is why bones show up strong (calcium being one of the heavier nuclei that’s common in the body). Contrast with ordinary MRI depends on the density of water and the speed with which magnetic polarization relaxes. It’s just a different process, so certain tissues with slight differences may show up better in one image technique than the other.
Once you get into fancier versions of MRI, using contrast agents, analyzing the spin-decay times, fMRI, and so forth, then you can get a great deal more informatoin than from CT. The reason is that it’s much easier to influence the amount and duration of magnetic spin polarization in your target than the amount of X-ray absorption. You can manipulate electrons (which control the magnetic shielding of the hydrogen nuclear spins doing the absorption) a lot easier than you can manipulate nuclei directly.
Contrast media is commonly used with X-ray CT as well. Indeed conventional X-ray imaging has used contrast media almost forever. But, as above, the answer is that - it depends.
Modern CT scanners are much faster, cheaper, and sophisticated than before. They are much cheaper than an MRI scanner. Plus they don’t have the difficult issues of trouble with magnetic materials. You can put one in the ER facility. My father (a now retired radiologist) was adamant that every car accident, or other physical trauma should have a CT head scan as a matter of course, to detect any sort of bleeding in the skull.
CT scanners can be more easily made with higher spatial resolution, and because of their higher speed can be used for gated imaging. You can gather images that are taken in sync with the heat beat or breath, and create high resolution images of the beating heart or lungs. Moving the phase even allows creation of movies.
MRI is particularly good in differentiating soft tissues. An X-ray image won’t show up cartilage damage, but an MRI of a knee will show up all the depressing damage, so sports medicine is a big user.
Pet peeve with terminology. CT means computerised tomograph. MRI scanners create tomographs (just a fancy word for a slice) as well, and are technically just as much a CT scanner as the X-ray version. CT scanners were once called CAT scanners, where the A stood for Axial - but since the advent of the helical scanner they are quite adept at creating arbitrary slices as well as axial ones.