When I started out snowboarding I never wore a helmet so now that I do wear one, it feels weird to have the straps on the outside.
Over the helmet doesn't sit as flush and snug against my face as under the helmet.
It works really well for me actually, I never need to take my goggles off and when I go for lunch or something, it's nice to be able to take off the helmet separate from the goggles without having them dangling on the clip.
IDK why people think under the helmet looks cooler though. I think over the helmet, showing off your whatever fancy goggle strap is cooler. (erego the giant O like Shaun white)