Solid stain on untreated exterior wood
I've done more digging, and this is what I've found:

Paint


Sits on top of the wood and is capable of forming a sheen 
Is good at keeping moisture out, but traps whatever moisture that does get in
Should be easier to clean food / drink spills off


Stain


Can't be applied over a previously painted surface (not a problem in my case).
Bonds to the wood more closely than paint because it actually soaks in
Allows moisture to leave the wood.
Does not form a sheen - leaves original texture of the wood exposed


In my application, I chose to go with paint. I applied two coats of exterior paint with "built in" primer.