jQuery(selector) vs. $(selector)

This is because of jQuery noConflict wrappers. It is certainly not an oversight by WordPress developers; but rather, a great little way to ensure we (developers) can hook our own javascript files or libraries without conflicting with other javascript code.

Be sure to read the Codex on jQuery noConflict Wrappers.

The usage pretty much depends on your intentions; and when you want the script to be executed (when DOM is fully constructed; as opposed to immediately).

It’s all explained very well on the codex page linked above.