What effect does vortexing have on a fluid sample that simple mechanical shaking does not?
In my experience, shaking and mixing have different "dead spaces". Supposed you had an eppendorf tube and you stirred it around with a pipet tip for thirty minutes. You would have great convective mixing in the radial direction but virtually no mixing in the Z-direction. Supposed you had a very viscous fluid like PEG. If you set the tube on a shaker for an hour it will virtually not mix. The vortexer provides a very different profile of mixing.

Instances where I wouldn't use the vortexer would be for sensitive objects like DNA and cells. 

(update)
We had some interesting suggestions from the FDA (since the FDA pays attention to these type of discussions). To get proper mixing, it is often not considered enough to merely swirl a tube a bunch. Their actual recommendation is to train technicians to perform a 90 degree angle arc with their forearm to generate proper force in mixing.