As for flat shooting, a 6.5-06 will launch the same bullet 200+ fps faster than the 6.5-284. That translates into flatter trajectory.
I run a 123gr Lapua Scenar at a hair over 3200fps out of my 6.5-06, using 59.0gr of H4831SC.
From working with other F-Class shooters, and having a quick look around, I see that the 6.5x284 guys are doing the same or in some cases better. Try 3405fps with a 120gr Sierra MatchKing and 63.0gr VihtaVuori N165:
With the 142gr Sierra MatchKings, the 6.5x284 benchrest guys are staying right around 3000fps:
That's about where my 6.5-06 runs the 139gr Lapua Scenars, too. I've also seen 140gr 6.5x284 loads out of a 26" Cooper run 3150fps over the chronograph.
Which only makes sense, because the 6.5-06 (69gr H2O) and 6.5x284 (65gr H2O) both have pretty close to the same case capacity, with the 6.5-06 gaining a whopping 4 grains of water. Other than that, one's long and skinny, the other's short and fat. The 6.5-06 runs at 50,000 CUP, and the 6.5x284 runs at 54,000 CUP. Both really need long actions to shine. Since I already run the 6.5-06, and the two are so close to each other in performance, I really don't have a great need for the 6.5x284 in my stable.