I agree with Kati's explanation the most.
Cute - usually something small, young and sweet-looking.
Pretty - superficially (on the surface) attractive, nothing extraordinary.
Beautiful - can be something physically or spiritually beautiful, and more of a deep ''goodness'' than the other two.
''She lived a pretty life.'' - I would think her life was about appearances, it was nice and proper but nothing out of the ordinary, maybe even kind of boring.
''She lived a beautiful life.'' A life that was meaningful, extraordinary and had value.
So cute, pretty and beautiful are really quite different in meaning, though all around the same theme.