Firefox has 2 search bars. Kinda. The address bar and the search box used to be configurable separately so that the search engine chosen for the address bar is irrelevant to the search engine set for the search bar. This behaviour has been disabled in FF23; the search engine selected in the search bar dictates the engine used in the address bar (the keyword.URL parameter doesn’t actually do anything anymore). So those of us who want 2 separate address bars are left out to dry.
UNTIL NOW. The Keyword Search plugin available here solves all your problems.