Native vs. Mobile First

Native Apps vs Web Apps have been debated to death. Every use case is different, let alone company, site, or task. Can the Mobile First theory help bring the two into a more comprehensive experience?

If the concept of needing a mobile app for mobile users vs. a mobile website with secondary menu or a Desktop link is the best way to cater experiences for context, I can't help but wonder why Mobile First strategy doesn't trump the idea of these two often different experiences for many sites. There are obviously many exceptions depending on the goal of the site where one or the other might best fit the task(s) at hand, but with this as a major take away from Neilsen mobile usability studies, it made me curious to check more into this and refresh myself on how Mobile First theory tackles this issue outside of the most obvious being resources.

I do realize that Mobile First as a strategy does not necessarily mean there aren't still multiple concepts - Native App vs Website or Web app. What I'm implying is that if you begin planning a web presence with that theory and you do it right, there should be an option from a presence standpoint on both but the experience shouldn't be that much different to push for one or the other - the use cases would be very similar - with the exception of offline use.