These are very similar, there will not be many situations where one is better than the other. "I will take the bus to work as usual/normal" is ok for casual English. I think 'as usual' works better when you are talking about something you do, an action, a habit or a routine, like taking the bus to work. I think most of the time 'as usual' is better.
'As normal' is perhaps better when you are talking about the state/condition of something, like "his temperature is the same as normal". So 'normal' often means something specific, like a temperature.
This is all just my opinion though, so I may be wrong.