Shaving soap has ingredients added that makes for thicker, denser, slicker lather over regular bar/bath soap.
Bar soap lather will be looser, more airy, with less cushion and glide and will break down quicker. One would be more likely to experience shaving irritation if using bar soap instead of a soap specifically designed to shave with because it is not as protective.
On another note, Mitchell's Wool Fat is the latest European shaving soap brand (others include Tabac, Geo. F. Trumper's, Truefitt & Hill, Haslinger, Speick) to move away from a tallow based formula.