In reply to jezb1:
Last winter Cogne had excellent conditions beyond Feb, which is not always the case. I've not been to either for a few years now, but in an 'average' winter (late Dec-early Feb) I would pick Cogne over LaGrave. Lots more routes of a moderate grade, easier to get to, more likely to be in. At LaGrave I've done both LePylon and Coliere du Ciel twice, on both visits there, as both times they were the only things in reasonable nick.
Big mountain feel? I think if some of those big La Grave routes on the left of the road as you drive in ever formed regularly, they would be incredible and to me looked much more daunting and alpine than anything in Cogne - though Cogne does have nice scenery in the background.
I'd def second/third the vote for LaBarme at Cogne, though it's not cheap. I stayed in cheap places in both Cogne and LaGrave (couldn't afford the Edelweiss) though the LaGrave place was like an abandoned mental hospital type hostel thing.
Cogne is famous as a x-c ski destination, all those flat trails, whereas LaGrave is equally famous as a bloody steep off-piste ski destination. I would go elsewhere to ski.