I second everything said here.
I will throw out two other considerations: solitude and navigation.
Some folks enjoy the social aspect of Whitney and don't mind the clutter of Trail Camp (TC), while others dislike the ramshackle feel of TC in peak season. Consultation is a way out of the way, but if you're familiar enough with the trail it is fairly easy to deviate from the Main Trail (MT) prior to reaching TC to reach Consultation. The combination of a little space and towering profiles of Mounts McAdie and Irvine might make Consultation the right choice for you.
If you do not have the best sense of direction I would strongly advise against camping at Consultation. Starting a hike in the wee hours or returning in the dark can make finding one's way back to the MT or back to your camp at Consultation challenging. The terrain there isn't dangerous per se, but getting off-route anywhere on Whitney can devolve into a dangerous situation.
