"(" or "\left(" parentheses?

There are in fact four important differences:

1. \left( ... \right) etc. scales according to the height and depth of its contents. This scaling is "dumb" in the sense that it will always take the full height and depth into account (how much of the expression is covered is controlled by \delimitershortfall and \delimiterfactor): for example, in \left( \rule{1cm}{1cm} \right), the parentheses reach far below the square. A more relevant example is \left( \sum_a^b \right) where the parentheses also cover the sum limits. The simple delimiters ( and ) and also the manually-sized delimiters \big( etc. don't scale.
2. \left ... \right forms a group: if you say \newlength\mylength \[ \left( \mylength=1cm \right) \the\mylength \] you get 0.0pt because the value was reset. More importantly, you cannot have line breaks inside \left ... \right groups, neither manual nor automatic ones, without special trickery. Any \left needs a matching \right.
3. Some characters produce different glyphs when being applied to \left etc. For example, < produces a less-than sign, while \left< produces an angle bracket. \big etc. use the same interpretation as \left (because they use \left internally). Technically, \left uses the delimiter code, while unadorned characters use the mathematical code.
4. The spacing is different. Technically, \left ... \right inserts an inner node, while ( inserts an opening node. This becomes visible in $\sin()$ vs. $\sin\left(\right). Therefore you can never simply replace ( by \left( and vice versa, you always have to check whether the spacing comes out right. An automatic solution to this issue is offered in Spacing around \left and \right, but the spacing within \left...\right can still be different as explained in this answer.

There is a "big" difference: \left( is used to adapt the size of the delimiters to the size of the characters in the math expression. You can try

\[ (\frac{x^2}{y^3}) \]

and

\[ \left(\frac{x^2}{y^3}\right) \]

and with \{

\left\{
  \begin{array}{rcr}
    x+2y & = & -1 \\
    -x+4y & = & 0 \\
  \end{array}
\right.

The delimiters can be : () [] | \| \{ \} or the dot .

I don't know what objective "best practice" there is, but I always use \left( when in math environments, and ( when not. This is because, as you probably know \left and \right macros scale things nicely, which isn't an issue in prose, but can be an issue when punctuation is used in other contexts.

I also use the former in inline-math environments, since scaling is an issue there as well.

Note that you will probably want to use \bigl[...\bigr] or \Bigl[...\Bigr] for larger square brackets (or parentheses) when you are grouping terms without fractions, especially across several lines.