AFAIK its only possible to do it via hardware capture of the output either with an external capture card (as seen in some youtube videos) or with debug drivers/card.
From a cursory look at them side by side it appears (probably not surprisingly given how AMD has marketed it) that VSR gives slightly superior edge filtering (anti-aliasing) while nVidia preserves intra-texture detail a bit better though its not cut and dried as in some situations AMD is sharper i.e. bright coloured text against a dark blue or black background is readable from slightly further away on AMD than nVidia.
Haven't checked but AFAIK FRAPS will suffer from the same issue - it will hook the frame buffer once the game has done rendering and before the drivers do their intercept for post processing.
You might be right on the latter, which means the only way to easily replicate it is to down-sample in photoshop and sharpen a little.