Why use a screwdriver when we have drill/drivers now?
If you can use a drill with a screwdriver bit to drive a screw, then most of the time, use it.

Here's when you might be smarter to use a screwdriver:


The screw is going into a pre-tapped (threaded) hole in metal, or plastic, or any soft material.
Always at least start such screws by hand -- so that you can avoid cross-threading them.
Putting screws into delicate assemblies: eyeglasses, electronics, medical prosthetics, etc.
When working in cramped spaces, a drill/driver often can't fit.  Screwdrivers come in flexible, stubby, right-angle, etc. varieties for such occasions.
For emergencies or odd jobs.  Multi-bit screwdriver tools are dirt-cheap now. I have one in all my vehicles. I keep one in my briefcase, one at my grandmother's house, etc.  You'd be surprised how often something needs to get screwed ;) and I certainly don't have dozens of cordless drills and chargers.

When the environment is likely to damage an electric device -- working in the rain, or in the mud under a house, or in a very gritty environment, etc. (Note that this is one more reason to make employees buy their own tools.)
When working with a material where finish is important.  A driver is much more likely to strip the screw interface, or to slip and mar the material being screwed, or to drive the screw too deep and crack or warp the surface of the material.
When driving slot-head screws.  These are notoriously hard to drive with a drill; the bit flies out of the slot too easily.
When a precise torque is needed.  (NEVER trust the torque setting on a motorized driver unless it is one of the industrial models made for that purpose and it has been recently calibrated.)
When the screw itself is delicate, such as plastic screws, soft metal screws, threaded wooden dowels, etc.
When working around volatile chemicals or in an explosive environment.
When working in a clean-room environment, screwdrivers are much less likely to contaminate and are easier to sanitize/sterilize.
When chemical or radiological contamination is an issue.  A plain screwdriver is much easier to decontaminate.
When the power has failed and/or the battery is dead (batteries also stop taking a charge after a while and need to be replaced).  Thanks to ratchet freak for the reminder.
When you want a nice vodka "kick" to your orange juice.