Inkscape's primary focus is
vector graphics.
Taking a screenshot is creating a raster copy of the content on your screen.
Raster graphics are built up from pixels which are supposed to display fine only at their original resolution -this case, on your screen. Current display systems have a smaller resolution than printing, which eventually should result in your print "smudgy" You could get way better results than that of your current print.
Like, contrary, if you made your pdf from vector content only -using the text tool for example- you could get crystal clear details, even if printed at a high resolution/large scale.