I keep them separate for two reasons: one privacy, and another preference:
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.
It prevents you from leaking data in cases where you like live search suggestions (like completions from your search engine based on partial text entered) but didn't want your search engine recording all the URLs you visit outside of search.
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.