1. If I'm interacting with the URL/address input, I'm either entering a full URL, or searching through my local history to jump to a previously-visited page (or, in Firefox, a tab I may have open on another device). I don't want that shipped off to my search provider for autocomplete suggestions.
2. If I'm interacting with the search input, I want autocomplete suggestions. Additionally, because it retains the value of the last term I searched for, I can use it as a shortcut to jump back to the results if I no longer have the tab handy.