A topic that comes up for many LaTeX users is how best to mark up math mode in sentences: inline math mode. LaTeX offers three (!) official ways to do that
$...$\(...\)\begin{math} ... \end{math}
The last version is clearly far too verbose for routine use, but the first and second approaches have a much less clear-cut division.
Plain TeX uses the $...$ construct exclusively, and that means many
experienced (La)TeX users simply use this without any further consideration.
There are good arguments in favour of the syntax, most obviously that this
switches directly into math mode (it uses the underlying TeX idea of category
codes with no macro expansion required). On the other hand, it lacks any
possiblity of matching begin and end points.
LaTeX’s \(...\) syntax was introduced by Lamport early in the development of
the format. Using separate begin and end marks means that is does allow error
detection in the editor, and it also is linked visually to LaTeX’s display math
\[...\] approach. (More on that below.)
So which one to use? Experience suggests that whilst Lamport made many good
decisions in the design of LaTeX’s input syntax, \(...\) wasn’t the best of
them. The number of times that pair-matching is helpful simply doesn’t compete
with the extra complexity of the input. Unlike $ symbols, parenthesis are
very common in math mode input, so using \(...\) is likely to lead to
confusion. At the same time, there’s no difference in the results between the
two syntax, so there isn’t a downside to using $..$. So I (and the majority
of the current LaTeX team) favour using $..$.
I think it’s important to contrast with (unnumbered) display math mode. There,
the LaTeX \[...\] syntax is the officially-supported approach, and the plain
TeX $$..$$ is not. For display math, there are significant differences in
what can be done using \[...\] compared to directly switching to TeX’s
display math mode using $$..$$, and so the situation is clear: use \[...\].