In general, in most places, it is said that using TeX defaults is the best. However, TeX does provide things like
\quad etc. When one should use them? Both in general and in math mode in particular.
The classic Mathematics into Type, by Ellen Swanson (the AMS has made a PDF copy available
TeX has built-in spacing, and most of the times it does an excellent job, so (most of the times) you don't need to add space manually.
The following document contains the summary of the rules given by Swanson and some examples; these examples also contain some cases which are not built-in and which require manual attention.
The compiled code looks like :
Addressing why, the table in the AMS' Short Math Guide for LaTeX, discussing spacing in math mode, provides a helpful approach. If we use the elements as intended then TeX will provide good (arguably, best) spacing. However, if the elements are used like an member of another class then adjustments to spacing may be needed:
Some of these adjustments are made automatically by TeX, e.g.
As others have pointed out, there are views on what changes to spacing improve the look, or ease of comprehension, of a document. Often an author will have to follow a house style but changes to the default should be considered carefully when a symbol is being used as TeX would expect. It should be noted, too, that best typographic practice varies between languages and over time.
One common case (in math mode) is with integrals:
is much better with
Another common case is with factorials: compare
Also, some people might argue that
Another interesting case is with
On the other hand, sometimes
(These examples are what I remember from The TeXbook, which I don't have at home; as far as I remember there are more of them there.)
In general, since both semantics and typesetting of math is so complex, it is actually quite often that you need fine-tuning like that.