Your new switch should make no difference over the old switch.
As others have said, the permanent live for the fan is not in the switch and should make no difference over the old switch.
Have you tried replacing the new switch, as maybe it is faulty and letting a small amount of voltage/current through in the off position what is enough for the fan to start.
Also if you have a multimeter, wire the new switch up like you did before and test the AC voltage between line and neutral at the bathroom light with the switch in the off position and see what you get. You should get 0v
As others have said, the permanent live for the fan is not in the switch and should make no difference over the old switch.
Have you tried replacing the new switch, as maybe it is faulty and letting a small amount of voltage/current through in the off position what is enough for the fan to start.
Also if you have a multimeter, wire the new switch up like you did before and test the AC voltage between line and neutral at the bathroom light with the switch in the off position and see what you get. You should get 0v