On the front end (i.e. in a Theme) don't enqueue jQuery or jQuery UI unless you actually need to use either of them. On the back end (i.e. in the Admin area), both will already be enqueued, since core uses them.
Whatever the case, if you use a core-bundled script, you should always use the version bundled with core, since core, Themes, and Plugins will all expect the core-bundled version to be registered.
That said, as of WordPress 3.2.1, not all jQuery UI plugins are included in core, so if you need to use a plugin that is not in the list on the
wp_enqueue_script() page, you'll need to bundle and register that plugin yourself.
As of WordPress 3.3, all jQuery UI plugins will be bundled/registered in core, and will thus be available to be enqueued.