bash for inline command

Others have pointed out the difference in syntax (basically, $() is slightly cleaner wrt nesting and escapes), but nobody's mentioned what I consider the more important difference: $() is much easier to read. It doesn't look like single-quotes (which mean something totally different), and the opening and closing delimiters are different, making it easier to visually distinguish its contents.

For your own scripts, this may not be really critical; code readability is good, but functionality is more important. But for anyone writing tutorials, sample code, stackoverflow answers, etc, readability is much more important. People will type single-quotes instead of backquotes when typing in examples, etc, and then get confused when it doesn't work as expected.

So for everyone writing examples on stackoverflow: please save your readers some trouble, and always use the $() form.

There's no difference except in "nestability":

The $() is nestable:

$ echo $(echo "hi" $(echo "there"))

while the `` is not.

$(...) and backticks are very similar. The only difference between the two is some details of what special characters are substituted in them; the Bash manual explains better than I could:

When the old-style backquote form of substitution is used, backslash retains its literal meaning except when followed by $, `, or . The first backquote not preceded by a backslash terminates the command substitution. When using the $(command) form, all characters between the parentheses make up the command; none are treated specially.

This makes it a bit easier to nest $(...), for instance. Besides that, though, there's no difference.