GoPro doesn't really go out of their way to advertise which modes have funky image artifacts and which don't (surprise, surprise). Though I'm actually surprised to see the apparent difference in sharpness in the video you linked, and would love to see test repeated on the same settings (or even 1440p-30).
Another possibility is if frame-blending is used to get 48fps down to 30fps in youtube, that will naturally result in less sharpness. That's not a drawback to the H3 at all, just a limitation on your choice of output (you can always shoot in 30fps native, use frame-dropping, or, if the H2 supports a 48fps mode, shoot in that as well and see how it fares with the same pulldown).
I'm still constantly impressed with videos people put up from the H2. If you shoot it and you like the output, there's no reason to go out and drop $400 on a new model. But I think it's worth being accurate about what differences and improvements do (and don't) actually exist.