Direct sunlight or sunlight reflected from normal surfaces isn't that bad for the eyes but "glare" coming off snow or other surface which cause specular polarization can really damage the cornea. Lenses with tint help reduce this but polarized lenses help even more. That's why the Innuit Snow goggles work: the slit in the goggle did a decent job of acting like a polarizing filter (google "single slit experiment") and also just reduced the amount of light which in turn, reduced incident UV.
How often are you on the snow? My son is over 140 days. I'm at 40. My wife at 30. No issues.
We have a lens for every condition and wear goggles always. If just on a bike, skateboard, longboard, hike, outside in general we use sunglasses that are polarized. I used to fish offshore alot and made polarized glasses a habit when outdoors.
Glare can cause strain on the eyes. Especially if you already have contacts or eye glasses. I would recommend polarized lenses, however if you don't want to spend the extra money Oakley makes great iridium lenses that still cut back on glare. Also most companies layer their polarization onto the lenses instead of injecting it into the lens. This can cause distortion and add to strain on the eyes. Oakley is one of the only companies i know if that injects the polarization into the lens, this is why they cost more.