What's the difference BETWEEN covered by something and covered with something?
They are close in meaning. If something is entirely covered, so that you can't see it at all, you could say either 'covered with' or 'covered by'. If the surface is not entirely covered, you can only say 'covered with' e.g the lawn was covered with flowers.
Can't I say I got up early this morning? Isn't it the same as I woke up early this morning?
Yes, you can say 'I got up early this morning', but it is NOT the same as 'I woke up early'. 'Wake up' means stop sleeping, 'get up' means leave your bed. If you woke up at 6 am then lay in bed reading for 3 hours, for example, in this situation you woke up early but you didn't get up early. They are completely different things.