it might be best if this is moved into the high end audio section...
i have used a very similar setup (kx->console->speakers). the biggest advantage is very simply, all the vst plugins that are availible. there are some free vst hosts, but i have not found any that are as easy to work with as console--i really like how i can 'see' the connection between different effects.
the best resource that i have found for vst plugins/hosts (both free and for $) is http://www.kvraudio.com/
there is no advantage from a processor standpoint--the plugins all take precious cpu resources to run, the real advantage is all the different plugins that can be added or rearranged to suit your fancy. there is also a very slim advantage that all the processing is done while the data is still in the digital realm-- to use most other sound processors(there are a couple processors that use a toslink cable, so this might not apply to those), the audio would be processed by the computer, converted to analog, and then redigitalized in the audio processor to be processed, and then reconverted back to analog to go to the amps.
by using the computer from the start to the finish of the process, the audio quality is slightly improved (compared to using multiple sequences of a/d--d/a convertors).