The fog beam is basically only foreground light: wide (ideally, though some aren't), very short reach/range, with a sharp cutoff at the top of the beam.
The auxiliary high beam (so-called "driving") is very much like a high beam headlamp: a strong central hot spot surrounded by less intense light.
The "driving" beam may be wide or narrow; terms like "pencil beam", "cornering beam", "combo beam", etc refer unofficially to varieties of aux high beam lamp.