The legal answer hinges on your location - that's the key thing that changes everything here. In most of Europe, daytime running lights are actually mandatory in many countries, and you can get fined for *not* having them on. North America is way different though; most US states and Canadian provinces don't require daytime headlights at all, and you won't get penalized for leaving them off. Some places have made them optional or only required in certain conditions.
About daytime running lights versus full headlights - they're usually treated as separate things legally. DRLs are those brighter front lights that many newer cars have, and they're often what satisfies the legal requirement in places that mandate daytime lighting. Full headlights are brighter and also illuminate your rear lights, which DRLs don't always do. So if your jurisdiction requires daytime lights, DRLs might be enough, but you'd want to check your local traffic code to be sure. Does that apply to where you drive, or are you asking about somewhere specific?