CT is a great screening exam because it is cheap, quick and more readily available. MRI is much more specific in diagnosing problems in the brain and has a lot more 'features' that can be added to aid in diagnosis. So generally, everyone gets a CT and then moves on to an MRI if there is an abnormality... or if they fail to find an abnormality when the clinical exam remains abnormal/severe. There are a lot of exceptions... like in little kids, you should always avoid radiation when possible and people will go straight to MRI.

To expand a bit on what Shinkei has said, and mostly this applies outside the brain as well:

CT is basically xrays in the round. You'll nominally use a radon transform (the actual approach is usually a back projection algorithm) to reproduce 3D position from a number of individual images. There are many good properties of this, it is relatively fast and cheap, it has very good spatial accuracy (e.g. a distance you measure on the image corresponds to a distance in the subject), but you have very little flexibility with tissue contrast. If an x-ray won't differentiate the features you are looking for, mostly you can only hope to add a contrast agent (typically expensive, fiddly, and more dose) where that is applicable.

By comparison, an MRI is more of a programmable imaging machine. You can get a lot of different contrasts by use of different pulse sequences, so you may be able to see features that are completely invisible in CT. On top of that, you can do interesting things with the time dimension and how things are changing while you image. This leads to approaches like functional MRI mapping which most people have heard of, but also diffusion and perfusion imaging. This can give you an idea of what is happening functionally, particularly in the brain. Downsides: time consuming (hence expensive), poor spatial accuracy, artifacts.

So for diagnostics or surgical planning, you get a lot more information from an MRI, but this also can take a lot more effort to interpret and understand.