A few things I've noticed between my 2003 Corolla and our 2007 Matrix:
The matrix is quieter inside
It's a smoother ride, yet seems stiffer at the same time, how did they pull that one off?
It seems to sway less while cornering. Now, I've been told the Corolla's sway is practically nonexistent, but when I got it I noticed an odd feeling when cornering that felt to me like the car was leaning, but I'm no expert so I might be interpreting the feeling wrong. After driving the Corolla for four years, I no longer notice it because I've gotten used to it, but in the Matrix I notice the *lack if that feeling, so there's something about the Corolla that makes me feel like it doesn't corner all that well. Not poorly, just... I'd give it a C or C- while the Matrix gets a B or B+.
It seems like the Matrix is easier to make a smooth start with, the Corolla sort of jumps ahead a little, this may be in part due to the Matrix's higher weight and lower horsepower, or the fact that the computer in the Corolla thinks I'm a leadfoot who always wants to drag race. ;-) It's not a problem, I just wish the throttle on the Corolla wasn't quite so sensitive in the first inch or so of travel.
The Matrix doesn't run the engine at 2500 RPM for a few minutes when you cold-start it (I'm not sure the exact RPM, but the Corolla immediately revs the engine and keeps it there on a cold start, I'm told this is to heat up the catalytic converter faster or something. I like polluting less, but not when it means revving the engine every single time it's cold-started. I'm sure it doesn't make much difference anyway, it seems pointless to me; and it's got to be hard on the engine.
The dashboard-located ventilation airflow directors in the M rotate 360 degrees and can all be shut off, whereas in the C they are the dual-shutter type, direction is more limited and directing to the side restricts airflow a bit, plus you can't turn off the two in the center.