'rest up' is a term often used when someone is ill or has had an injury, and they need some time to recover from this. For example, a sportsperson who has strained a muscle may need to rest up before competing again. 'rest' is a far more general word. You rest or 'have a rest' (GB) (or maybe 'take a rest' in US Eng) when you are tired.

I think there is no significant difference. "rest up" is more colloquial but means the same thing. I would not worry about this and go on to some other English words, phrases, idioms, etc.

There is very little difference between the two. The only real difference I can think of is that you can rest a little while standing whereas rest up implies lying or sitting with your feet up.

Thanks guys