Stále is used for a continued action, and means that the actor doesn't end doing it. In this context it means that he seemingly never wakes up. A synonym is pořád.
Vždy is used for time. For example, "vždy můžeš začít". An emphasised version would be "vždycky".
A useful way to think about it is "vždy"="always" and "stále"="all the time".