According to http://www.ghacks.net/2015/12/16/how-to-restore-classic-search-in-firefox-43/, you can use Classic Theme Restorer add-on.

browser.search.showOneOffButtons was a temporary pref starting in Firefox 34. Many of us expected that pref to be gone by Firefox 38 or 39; so you can look at it like this - we got 3 to 4 more versions of life out of it than what would be normal these days. The underlying code that supported that preference is gone as of Firefox 43. I guess that it took Mozilla longer than initially expected to 'perfect' that new feature.

As already mentioned ... https://addons.mozilla.org/en-US/firefox/addon/classicthemerestorer/ has an option for Old search (experimental) in CTR Options > General UI (1)

Don't worry about that (experimental) tag, it seems to work fine for me with Firefox 43 although I mostly use the ESR 38 version of Firefox for good part of my day.

Thanks! I've installed the add-on, but don't know how to change the behavior of the search button. It's still the same. This one piece of functionality is the primary reason I use Firefox over Chrome.

Was much easier before and I don't like that addon, but I guess I have no alternatives, right?

Lev,

I don't like that CTR extension either, I hated the UI it is based upon when that UI was the 'current' Firefox UI. But until someone packages the old Search Bar code in its own extension there isn't a real solution to this dilemma.

I am using the Firefox 38 ESR version currently until I get up-to-speed with the SeaMonkey browser. The predecessor to SeaMonkey (the Mozilla Suite) begat Phoenix in 2002 which became what we now know as Firefox a few years later.