Shanahan:
In my mind, Shanahan is the "big picture" guy. He, with input from all the hockey minds in the organization, basically decides the direction that the team will go, the type of players that should be on the team, etc. Essentially, he creates the "mission statement" that guides the hockey department in doing their jobs.
Nonis:
It's Dave Nonis' job to manage the roster. He's the one who negotiates contracts (or oversees them being negotiated), talks to other GMs about deals, scans the waiver wires for pick-ups, etc. He uses the "mission statement" to guide him in doing his job.
Nonis at the Draft:
As for Nonis looking somber at the draft, that may be nonsense. Nonis is a somber-looking guy most of the time. He just doesn't smile a lot. I suspect people are reading far more into this than what is there.
Thoughts
Probably, like many business relationships, the relationship between Nonis and Shanahan will be unique to the two men. There may be blending of roles at times, for example, when Shanahan has a more useful relationship with a team than Nonis, he may be more involved. There will also be areas where, yes, Shanahan technically has final say, but he trusts Nonis in those areas, so he happily rubber stamps Nonis' decisions. That's certainly the way it often works in other business situations. For example, while my manager technically has final say on all projects that our department handles, there are some projects where she trusts me to handle the project management duties and simply rubber stamps my decisions.
Personally, I wonder if some fans simply like to be able to have one specific person to blame for all decisions. It makes their angry Internet rants easier.