I currently have both, a SE Saloon 2004 Auto and 2005 Exi Prosmatec. Here is my take on it.
Civic Pros : Better Exterior, Road Grip, Braking,
Corolla Pros : Better Interior, Soft Suspension hence comfortable, good Road clearance, strong body and specially good bumpers + Less Engine noise inside cabin
Civics suspension seems very hard to me, Plus its bumper is literally sensitive. Also, at 3000 rpm, the sound inside the cabin is very annoying.
I don't know if its just my car or is it the same issue with all : I feel SE Saloons transmission much better in shifting and response than Exi Prosmatec.
There is no definite word on which is the better car rather both have somethings good and bad than the others.