To me mounting the print and under an over-mat sort of elevates its status a little bit. I feel that I am able to view the picture differently in that I see the whole picture better.
When I hold a print in my hands, which I love to do, I always wonder what it'll look like behind an over-mat.
My preferred sizes are 6x8" or 9x12" (same aspect ratio of 3:4) with 1" border for rectangular pictures, and square I print 6x6" on 8x10" paper, 8x8" on 11x14" paper, and 13x13" on 16x20" paper. I absolutely love holding these prints in my hands, study them, feel them. You mentioned how the print feels in your hands - this is why I love matte paper for example.

In summary I think putting the print behind a window mat or a frame will distance it from me, allowing me to view it from a 'big picture' perspective where I pay much less attention to the details and appreciate it more from a subject matter perspective, as well as visual impact. But it also robs me of the intimate viewing experience and the wonderful sensation of holding the prints in my (freshly washed) hands.
My goal is, however, always to get the prints I love at least behind a window mat. To me that's sort of its final destination.

- Thomas