Can a regular CT scan in the ER miss things going on in or around the brain?

Possibly. Ct can be performed in a variety of ways to demostrate abnormalities f the structures in the skull. A standard non- enhanced study does not show certain changes in the brain or blood vesssels. If your doctor is concerned about these then additional imaging with tailored ct or MRI might be appropriate.
Diagnostic limits. In the er, the docs work to prove you are stable or unstable. The er ct scan will show major problems. Mri and ct angio show more, but not done from er. Absent major findings or problems, the er doc will allow you to finish your work up as an outpatient. Meaning follow up with your regular doc who works to maintain your health.
Resolution. I'm not a radiologist or a neurosurgeon, but in my field we use ct vs. Mri for pituitary tumors based on resolution of the study: a MRI is reliable for resolving small tumors down to 1 mm. Ct is also highly reliable, but the resolution is a little less.