I would have to argue against similarity when looking critically at what is expanded or broadened.
"broaden one's horizons" could be considered to have to do more with experiences, physical knowledge of the world, for example working lots of different jobs will broaden your horizons in ability and understanding.
"expand one's outlook" could be considered more of a metaphysical experience. A philosophical expansion of how one thinks, understands, or sees the world - Solipsism vs Existentialism, Hedonism vs say Puritanism or Asceticism and so on. This has less to do with worldly knowledge (as in expanding your physical ability and experiences) and more to do with philosophical knowledge (expanding mental understanding), but they are closely related as the physical world shapes our minds and the world is also shaped by our minds. But this is just my opinion...