When should screws be used for general wood framing?
The short answer is that, for residential framing, the only situations that call for screws are:


Where building hardware specifies screw fasteners
Where squeaks are a concern and adhesive would be impractical
Where access is limited with respect to hammers or air nailers
Where the need for later non-destructive removal is expected


Everything else is usually done with appropriately-sized nails or staples, some of which should be adhesive-coated, ring-shanked, or otherwise special. 

I won't endeavor to list all appropriate fasteners and their use-cases here, as that would be an inappropriately broad topic for this site. The most common pneumatic fasteners in my area are ~3" cement-coated framing nails, ~2-3/8" ring-shank nails, and 1-1/2" to 2" cement-coated narrow-crown staples. 

Construction adhesive is appropriate for reducing noise and making structures feel more solid, as in staircases. It is not, however, rated for weather resistance scenarios. Mechanical fasteners and building hardware fills that role.