The difference between "quite" and "enough"

Is there any difference between "quite" and "enough"? Please suggest the proper usage.

I'm not quite happy.

vs

I'm not happy enough.

Quite - a quantifier, specifying a finite, discriminate amount. In the case:

I'm not quite happy.

You are supposing that there exists some numeric scale of happiness, and that the measure of happiness which you have is less than that needed to be qualify as "happy". In this phrasing, there is no overlying sense of the commitment to seeking happiness; the phrase is intended instead to provide a sense of scale of the happiness of the person. Use of the word quite here could be done in the following manner:

I experienced five problems in the last two days. I am only happy when I have experienced less than two problems in the last two days. I'm not quite happy.

Whereas:

Enough - sufficient to a degree; qualitative.

I'm not happy enough.

This phrasing has a relationship much more with the degree to which a person is committed to seeking a goal. No amount of happiness is specified; moreover, the use of "enough" is intended to note primarily the discrepancy between where the author is and where the author desires to be. The following use of "enough" may be helpful:

I've been having problems, and problems make me sad.
It's not that I'm always in a dour mood. I'm not happy enough.

Not quite means ‘not completely’. Not . . . enough means ‘to an insufficient degree’.

Your examples are unlikely sentences, because happiness is not generally talked about in that kind of way. A better example of the use of not quite would be ‘I haven’t quite finished’, and a better example of the use of not enough would be ‘ ‘I haven’t studied enough.’