Whenever you are crossing a Schengen border, you must de jure be in possession of a valid passport or accepted national ID card.
In many countries (I believe Austria to be one) you need to be able to identify yourself (passport or EU member state national ID card) if the police asks you to.
If you are driving, of course you must have a valid licence on you ...
Most European countries require, upon demand, that you identify yourself and in cases of foreigners their legal status.
For this your passport and visa is required.
A driver's licence is for driving and is not considered an identification of nationality and legal status.