Both display math $$...$$ and inline math $...$ is valid in LaTeX. In fact, their accepted counterparts \[...\] and \(...\) are defined in terms of the former. However, \[...\] just adds a little flair in term of spacing around the environment by performing some checks. The use of \(...\) is almost exactly like using $...$. To see why this is the case, view their definitions in latex.ltx:
\def\({\relax\ifmmode\@badmath\else$\fi}
\def\){\relax\ifmmode\ifinner$\else\@badmath\fi\else \@badmath\fi}
Of course, you can also use more contextual phrases to do the same thing: \math...\endmath is an environment pair (similarly, \begin{math}...\end{math}) that is defined as equivalent to \(...\), and also \displaymath...\enddisplaymath (similarly, \begin{displaymath}...\end{displaymath}):
\let\math=\(
\let\endmath=\)
\def\displaymath{\[}
\def\enddisplaymath{\]\@ignoretrue}
In answer to your question: Both forms are usable by amsmath, but it actually only supports the use of \[...\], since it redefines these to mean something different:
\DeclareRobustCommand{\[}{\begin{equation*}}
\DeclareRobustCommand{\]}{\end{equation*}}
Convention, over the years, has shifted to using \[...\] for display math and a mixture of using \(...\) or $...$ for inline math.
$…$may no longer be supported. So in the interest of future compatibility,\(…\)should be preferred. – Konrad Rudolph Oct 06 '11 at 09:48