It applies to all winter sports countries that not driving with snow tyres may lead to being deemed (co-)liable for accidents because a driver using normal tyres cannot respond as well to unexpected situations.
Germany and now Austria as well go a little bit further than this: Cars (including foreign cars) must have winter equipment when driven under winter conditions. In particular, snow tyres and sufficient screen washer liquid that are suitable for winter conditions are meant with winter equipment.
This obligation also applies to foreign cars. In addition of incurring a fine (min €50,- max €5000,-), not using snow tyres can lead to being deemed (co-)liable for accidents because the driver driving a car with normal tyres cannot respond as well to unexpected situations. Without snow tyres you will, after all, have less grip on the road and the braking path will be longer under winter conditions.
If snow tyres have been installed, you must keep to the prescribed maximum speed for this type of tyre. If this speed is lower than that of your car, this maximum speed must be indicted with a label that is visible to you.
Why snow chains?
In some places in Austria snow chains are mandatory, indicated by a road sign. Don't bring your car, yourself and/or your family unnecessarily at risk because you did not want to invest in some good and easy to install snow chains.
Also on the way back home driving down on the mountain pass, snow chains can be vital in bad weather and road conditions.