Type R is what Honda uses to badge their race inspired platforms. The NSX-R, Integra Type R, Civic Type R and the Accord Type R.

All these cars take their existing platforms and undergo major changes such as supension, chassis, engine as well as interior upgrades. If you put a Integra GSR next to an Integra Type R they look the same externally but differ a lot internally. The Honda R cars are not common and are usually in limited production too.

The Type-S platforms are made for the mass market. They are usually the top of the line trim for that car platform. (CL Type S, TL (Saber/Inspire) Type-S, RSX Type S...etc. etc.) These cars were not made in limited production.