To clear something up is to communicate effectively in order to resolve a misunderstanding or misapprehension.
Ex: Ronda called the office to clear up some misunderstandings they had about her project. She explained the meanings of the notes that went with the presentation and answered their questions.
Ex: After the presentation, the speaker lets the audience ask questions to clear up any misunderstandings they might have. The project is quite complex and it's especially difficult when it's being presented in a language that is not your own.
If a medical condition like a rash goes away, it also clears up.
Ex: I was pretty sick when I first got to Florida and I thought my vacation would be ruined. But I took some natural medicines that my aunt had and my cold cleared right up.
Ex: Poison ivy gives a pretty nasty rash, but if you just take a bath in oatmeal and rest for a couple of days, it should clear up. Then you'll be able to go back to work like nothing happened.
To tidy up is to clean a mess by putting things back in their rightful place. Cleaning may mean using a broom and mop and chemicals, but tidying up is a bit faster and more about putting things back in order.
Ex: Dave really hates tidying up his flat, so it's often rather messy. There isn't a lot of food or dirt around, but clothes and books are just lying around the floor all the time.
Ex: The angry mother told her children that they all had to tidy up their messy rooms before they could have any ice cream. There were toys everywhere in there.
Tidy is when something is clean and orderly. Sometimes we use tidy to mean large or generous, for example with the words bundle or salary.
Ex: The kitchen is very tidy. I have put all the dishes away and organised the plates and the pots and pans according to size and colour.
Ex: Chris started working for a new company last year and apparently he earns a pretty tidy salary. He's been able to afford holidays in Thailand this winter and Turkey in the spring.