I have owned both. It was not a 05 Type-S but a 04 with a K20a2. They are both great cars. It boils down to personal opinion. The Si has navi as a option which is nice. It also has a ipod hook up. Tier dash is not for everyone but I like it. It looks advanced to me. The stereo in the Si is way better then the bose in the Acura. Now as for performance my 06 Si with just a CAI ran 14.7 where as for my Acura with CAI and catback I ran around the same. So as you can see performance it close. The civic is better in the corners bar none compared to the Acura. As for the 3rd gear issue and rev hang from the drive by wire are all fixed under warranty due to the recall. I had the 3rd gear issue on my RSX so it is not limited to the Si. As for looks you decide.
My Current 08 Mugen Si

My Old 04 Type-S A-Spec