As for the dream meaning, songe and dream are the same - only songe is more litterary. If you say "cette nuit, j'ai fait un songe", one would expect something more unusual than a simple dream... e.g., The French Bible relates le songe de Jacob, not le rêve de...
Now, songer also means to think, to consider : Quand j'y songe, traducteur est un beau métier
Je songe à abandonner mon métier de traducteur
This was my doubt as well, dreaming while being asleep versus awake (day-dreaming). My dictionary indicated the meaning is identical, but I wondered if, as JeanDeSponde indicated, there are certain situations where one word is more appropriate than the other.