From a user standpoint, I suggest ordering the Search Existing menu in the following order: Google Search (keywords only), Simple Search (status, product and keywords), and Advanced (all fieds).
Google Search should definitely appear last. Two reasons: 1) Google search only indexes publicly available bugs. There are hundreds of thousands of private bugs in Red Hat Bugzilla. 2) Google doesn't search bugs in real time, and therefore there is a lag between when a bug is updated and when Google indexes it.