Please explain to me the difference between them . When should i use them and so on .
What's the difference between 写下 and 记下？
记下 is recording. It is used for objective things, such as an event. When someone 记下 something, the person is an observer.
写下 is to put something in writing. 写下 is broader in meaning than 记下 and includes the meaning of 记下.
写下 may have two special meanings. (1) To write some ideas and plans in your mind that are not as realistic as the event or things that exist objectively. (2) To communicate with others using a written language. Such as 写一个总结报告.
In literal meaning, 写下 means "write down" and 记下 means "jot down". They have very slight differences in meaning which makes them basically the same in most of the contexts.
If you have to ask, the former simply tells people to write something, and the latter emphasises recording or logging something. Like we usually say "Write down the answer on the box given." instead of "Jot down the answer in the box given.". The latter emphasises logging the thing instead of simply writing.
CN: 写下：代表利用工具，如圆珠笔等将信息书写在纸张上。 记下：虽然也有记录信息的意思。然而还有引申出遵守意思，例如老师教育学生如何解决问题，学生向自己强调将来会按照老师的方式来解决问题。 EN: Writing: means using tools, such as ballpoint pens, to write information on paper. Note down: although it also has the meaning of recording information. However, the meaning of compliance is also extended, for example, the teacher teaches the student how to solve the problem, and the student emphasizes to himself that he will solve the problem according to the teacher's way in the future.