I'd say a programmer is to a software engineer as a handyman is to an architect. Can a handyman do the "dirty work" of most projects? Sure. Can they design a skyscraper to spec, tell you exactly what to expect in terms of resources and timeline, and roll with the punches of logistics and competing priorities? Probably not, unless they were already an architect just doing the job of a handyman.

That being said, as someone with a software engineering degree, I can safely say that most companies don't need engineers and can get by just fine with programmers. Anyway, in this industry there are many architects working as handymen and vice versa.