These are different type of enhancement. VAD, sometimes called squelch, is about suppressing low loudness sections, in the time domain. Although I couldn’t quickly find details of the Speex donoise algorithm which is what Asterisk accesses, it is likely to create adaptive filters, which remove parts of the frequency domain not currently carrying speech information.
As far as I can quickly tell, Asterisk doesn’t access Speex’s VAD features for general use.
I don’t know what the Cisco phones do, but, in principle, a phone can use multiple microphones, to distinguish background and foreground sounds, but this information is lost by the time it reaches the PABX.
Speech noise reduction is one of the big problems that the hearing aid industry would like to solve. It isn’t easy.