Simple. No. ANZ Private does not discount on rates. You'll get the standard 0.70% and nothing more.
They have more experienced bankers and credit is better (easier to get the loan approved). However, it very much depends on the person you deal with - like anywhere there are good and bad people. The 'Private' name doesn't guarantee consistency.
My advice is if you call your bank/banker regularly (say 3 times a week or something like that), you'll get value out of Private. However, if you rarely call on your bank, other channels will give you a lower cost.