You need to retrieve the Trouble Codes (count CEL flashes as explained in a Service/Repair manual like Haynes) in order to see what exactly is causing the CEL to stay lit. Most likely, it's emmissions-related.
Yes, if the problem that initially turned on the CEL was fixed, driving it around will let the ECU "discover" that said problem's been fixed and it will turn off the CEL...it's probably not going off because the problem hasn't been fixed yet. If you were to turn the CEL off manually but didn't fix the problem, it would just come back on again.
John D.