It's a question of register - they mean the same, but (as is the case with most phrasal verbs) 'tied up' is more informal.

Yes, you can say either.

It is true, you can say both and mean the exact same thing. It is a choice between using it in a formal setting or an informal setting. You would sound more proper and would use "occupied" when using it in the professional sense with someone who you may not know. "Tied up" is informal and also a popular English expression meaning busy doing other things and would be used with people you know. It would be more common to say that you are "tied up all day" and perhaps make you sound more fluent with English ;)