What's the difference between "priority" and "precedence" and how to use them?
How to distinguish them while expressing that something should be done first? I think generally saying "it takes/needs precedence" or "it should be given priority" both works. I'm kind of confused.

Question

Answer

"take priority (over sth)" and "take precedence (over sth)" are close synonyms. "precedence" is more formal and less common than "priority".  
There is a small difference in meaning: priority refers to importance, whereas precedence refers to sequence of time.  Often, the difference is only one of emphasis.
Other verbs like "give" and "have" can be used similarly.