I guess what I'm saying is that the midi path in a situation like this is normally:

Keyboard -> Midi In on interface -> DAW software -> Midi Out on Interface -> sound module.

Sometimes the interface, or the DAW, are designed to forward the incoming notes thru to the out port. Sometimes they are not.

There should be a way to configure that in Reaper. (In Logic Audio, it's a shiny button labeled "THRU", and it's off by default).

One reason it might be off by default is it's assumed that you're wanting to use Virtual instruments instead of external instruments to make the sound. And some people when they plug in their systems plug in both the IN and OUT cables to the same keyboard thus getting doubled notes because of a loopbacked THRU.

Cakewalk Home Studio might simply have had THRU enabled by default and you never had to change it. I'm saying maybe Reaper is different.