Of the two, I'd prefer the 6.5-06.
I have a .260, and also have the ballistic twin on the 6.5-06, the 6.5-284 Norma.
Given a choice between the 6.5-06 and the 6.5-284 Norma, I prefer the 6.5-284. Mine is built on a long action, and I'm 300 rounds into my second barrel. Factory cases, factory dies, abundant data, abundant collective experience. . . Since it's based on a short-action case, reaching the lands in a standard chamber is no problem with ammo loaded to a Rem 700 LA, even with Berger 140s. I get around 3,000 fps with 140s and H4831SC. Barrel life will be similar between the 6.5-06 & 6.5-284, and I had sub-minute accuracy from mine to 1,700 rounds, and then shot it to 2,037 with 'hunting accuracy' of less than 1.5 moa.
The .260 delivers better barrel life, equivalent accuracy in equivalent rifles (both of my 6.5s are GAPs), but 200 fps less with similar bullets.