hook_user_insert() vs hook_user()
The simple answer is that hook_user is for Drupal 6 and hook_user_insert is for Drupal 7.

Most (all?) of the API functions in Drupal 6 and earlier that accepted an $op parameter got split out into individual functions in Drupal 7.  This happened with hook_block, hook_nodeapi, and some others.

For Drupal 7, the full set of user hooks is described at user.api.php.