Asphyxia is the major one followed by trauma. Hypothermia is rare. This is due to the excellent insulating properties of snow.
Real cooling starts when the victim is dug out and exposed to the elements. Hypothermia is always, always, always an issue in wilderness emergency medicine, but is not what kills avalanche victims.
Taken out of this recent review of avalanche victim resuscitation by none other than mountain rescue medical sensei Hermann Brugger & co. Essential reading for anyone doing mountain medicine or wilderness medicine.