<Then I put that same lens on a D800 and frame the shot (by moving my position relative to the subject) so I get the same area that I saw with the D300. Now, take those two images and resize them (without any cropping) to, say 1200 pixels horizontally or print them both at, say 10 inches across. Will I see any difference? If yes, why?>
You will of course have a change in perspective through moving close, so they will not be exactly the same.
Downsizing to 1200 pixels means you are resizing 6 pixels to 1 on the D800 and 3.5 pixels to 1 on the D300. That's a big advantage for the D800 and it will show a lot more detail. Some of the sample images look almost too sharp to be real because of this effect. The D800 will also have significantly more dynamic range and color than the D300.
Now this is a bit unfair as the D300 remains a very good camera and the equal of almost every published image you have seen over the past few years. The D800 just moves things to a new level.