What can a CT scan of the brain catch that a brain MRI scan cannot?

Also, very fast! Agree with excellent answer above. Additionally, ct scans have the advantage of being much faster than MRI scans, and they are much more widely available- if you come into a small hospital's er at 3 in the morning, they will likely be able to do a ct immediately, while an MRI might wait until the next morning (if it is available at all!).
Bright stuff !! A ct of the brain is really good at visualizing highly calcified objects (think x-rays) such as bone. Blood can be visualized in areas where it should not be like the subarachnoid space. Ct's are great for visualizing acute bleeds such as might result from trauma.
Depends. The most common reason for an immediate CT scan in the Emergency Room is to evaluate for abnormal bleeding onto the brain. Time is critical for a patient with bleeding into the brain, and a CT scan may take less than one minute to perform, while an MRI may take up to 30 minutes. Bone imaging with CT is also better than MRI at times. MRI imaging is better than CT of the brain for many diseases.
Not much. Perhaps detecting a skull fracture or bony abnormality (like in the temporal bone). Mri is more sensitive than ct in detecting stroke, most hemorrhage and tumors. That being said, ct is often done first because it is fast and available 24/7.