TO WAKE UP
=to stop sleeping and open your eyes.
When your alarm clock goes off in the morning you wake up because you are no longer sleeping.
You can wake up naturally (you body doesn't need to sleep any more) or something or someone can wake you up such as an alarm clock or a friend (or dog) jumping on your bed.
TO GET UP
=to get out of bed (usually to start your day).
Get up can also mean: To go from a lying position onto your feet.