It depends... do you want comfort and amenities in a relatively quick sports coupe or do you want a balls-out purpose-built track car?
The Integra Type R is faster, geared shorter, handles better and is lighter than the RSX Type S. The downside? She's buzzy and noisy as hell (stripped of sound deadening material), no cruise control, sunroof or hell anything but an A/C in latter models (option in 97 and 98) and a radio, stiff suspension and seats so it's not as fun as a daily driver and, of course, your insurance will be higher than on a Type S.
Buuuuut, if you're like me and enjoy a balls-out purpose-built track car, she's the best-handling FWD car ever made (let's ignore the new Mazdaspeed Protege and possibly the SVT Focus

h: ) and has a very stout engine and drivetrain

I bought her to compete in AutoX and around road courses when time and money permit (with occassional jaunts to the drag strip) and she's perfect in those applications. Which is why I bought a 2000 Type R last summer
Basically, do you want to take the car to a track where speed and handling matter? Don't just consider the limited availability and reputation of the Type R. If you just need a car to go get groceries, go to class, etc., spend the extra $3k and buy a Type S.