Well, for the default Google searchbox the advantage is:

  Ctrl-K <search phrase> ENTER

versus

  grab mouse .. point ... click ... move hand back to keyboard ... type
search phrase, ENTER

I never used any of the secondary search bars.  Unless I can define a
shortcut to access hoogle quickly there's no advantage.

To search Hoogle I just type Ctrl-K F4 h <search phrase> ENTER. Once you're in the search box, F4 (or alt-up/alt-down) let you select a particular search engine.

-Brent