Doubts

I know that we use 'do' like auxiliary verb for negative sentences.
For instance:
I don't want more pills.

But, can we tell any difference in this statement?
I want no more pills.

Is it correct?

No real difference, you're correct. The 2nd sounds a little more definitive, though.

I  believe 'no' is used differently in English than in Spanish (I could be wrong), in that, for instance, saying "I'm no teacher" is a little more intense than saying "I am not a teacher".

Typically, 'no' means 'not any', 'not at all', whereas 'not' means 'for the most part not'.

Doubts

I know that we use 'do' like auxiliary verb for negative sentences.

For instance:
I don't want more pills.

But, can we tell say?
I want no more pills.

Is it correct?

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I think yes its correct. 

Doubts

I know that we use 'do' like auxiliary verb for negative sentences.
For instance:
I don't want more pills.

But, can we say  tell?
I want no more pills. This is considered uneducated English, spoken by the lower classes.  The correct and formal way is, "I do not want any more pills".

Is it correct?