If $r$ and $s$ are multiplicatively independent integers, then it is known that there are (lots of) numbers that are normal to base $r$ but not to base $s$. I think Wolfgang Schmidt may have proved this, and there is also work by Andy Pollington.
EDIT: a reference is Wolfgang M. Schmidt, Über die Normalität von Zahlen zu verschiedenen Basen, Acta Arith. 7 1961/1962 299–309, MR0140482 (25 #3902). This builds on earlier work of Cassels.