What is the Difference between don't and do not-where we use these words?

'Don't 'and 'Do not' both mean the same thing. 'Don't' is a contraction (shortened form) of the phrase 'do not'. It is fine to use either. For example, you can say the following, and it will be correct -
1. Do not touch that - or - don't touch that
2. Don't talk to me like that - or - do not talk to me like that
3. Don't you have anything better to do? - or - do not you have anything better to do?

Here are examples of other contracted words -
1. Madam - ma'am
2. Is not - isn't
3. I am - I'm
4. Where is - where's
5. How would - how'd
6. Would have - would've
7. You will - you'll

Do not is the same as that of don't. You can use either of them in a similar manner. There will not be any difference in the meaning. There are no special rules regarding the usage of these words. You are free to use any one of these two.
I do not want or I don't want.
I don't know or I do not know.
I do not like or I don't like
are some of the examples came to my mind.

As member's have sid, both are correct provided when and how you use it. Don't is the diminution of Do not and act as auxiliary verbs. Many other word have such diminution.

It can be used to make a negative statement or when using questions (question tag):
In English, don't is used when speaking in the first and second person Singular/plural and the third person plural ("I," "you," "we," and "they").

1. Negative Sentence:
I don't like tea.
You don't want to do that.
We don't want to go out today.
They don't have to answer.

2. Question Tag:
You want to call her, don't you?
Don't they understand?

In English language with time certain modifications took place due to imagination of the writers and literary persons. These changes were not the drastic ones but in the nature of writing things in short form or abbreviating a set of certain words and things like that. Sometimes the composers of prose or poetry preferred to write what we pronounce rather than what actually is there as per spellings. These are the reasons for culmination of a set of word to a single word with apostrophe.

The evaluation of 'don't' from 'do not' is one such example. Other examples are -
'can't' from 'can not'.
'It's' from 'It is'.
'I've' from 'I have'.

I will like to quote a few lines from the lyric from the album 'A fever dream' year 2017, which shows use of this type of words -

It's up to me, it's up to me, it's up to me
I've gotta try it again
It's up to me, it's up to me, it's up to me
I've gotta figure it out
It's up to me, it's up to me, it's up to me
I said, I said
Help me I can't do the thing you want
Can't do the thing you want

Don't is the shorter form of 'Do not'. In earlier days, at the time of writing, people used 'Do not' and while speaking, used to say 'Don't'. However, nowadays there is no such distinction and both the forms are used while speaking or writing.

This is really a good question.
The first thing is both don't and do not are same but their usage are different.
For example if we are writing a letter to our boss then it should contain "do" Or "do not " In formal letters, articles, submissions.
While in informal submissions we often use "don't".
So don't is normally used in rough and informal language.