What do you mean by "the voltage at the positive input" (where is it measured), and what do you get?
There are two switching points in that circuit, 3.33V and 3.19V (13.3V and 12.8V at the battery terminal).
For high side switching, you should use a P-channel MOSFET.
Consider buying a battery protection circuit, which is cheap and will protect from overcharge, overcurrent and overdischarge.