As someone who did work experience in a company that had a static safe room (tables earthed, floors earths and you are earthed) it is earthing and not to keep you at the same potential voltage. The problem comes when you build up static and that is discharged into the component in question, this involves a small current but high voltage that fries the chips hence the reason why tyhe strap is nconnected to earth (and why croc clip based straps should be connected to part of the computers case as that is earthed through the power supply and any potetialy harmless voltages sent to earth).
As for brush type, I don't know as we just had whatever was available (poorly funded course with computers found in a cupboard! and computers chucked out from Bournemouth Uni. only got proper computers (cheap dell laptops) at the end)