There are two sorts of "autopilots", and it is important to make a distinction between the two. One is for the behaviour of the aircraft around its Centre of Gravity (CoG), the other one is for defining the path of the CoG.
The Inner Loop autopilot: behaviour around CoG, or the aircraft attitude control. This autopilot should not be called autopilot since its characteristics are actually shielded from the pilot: it does its job without showing what it is doing. It only moves the control surfaces (elevator, ailerons) or helicopter swash plate, and not the flying controls (stick, wheel). To the pilot, the system is invisible and just results in a stable platform, in the case of the A320 and B777 a platform with envelope protection that does not allow for the aeroplane to enter a situation that would be dangerous. In helicopters this system is referred to as Stability Augmentation System (SAS).
The Outer Loop autopilot: defining the path of the CoG, which is ultimately what pilots are paid for :). They are in control of the flight path and administer this control by deflecting the flying controls (stick, pedals). The pilot can delegate this control to a feedback system that provides inputs to the same flying controls, via an actuator with a slip clutch. With an old fashioned mechanical linkage, the actuator moves the whole circuit from stick to surfaces (or swash plate in a helicopter); the pilot can override this input by applying a force greater than the override force of the slip clutch.
So normally, by design, the Outer Loop autopilot shows the pilots what it is doing by moving the flight path input devices for them, in a way that is transparent and can be intuitively overridden by applying a high force. It is this bit where the A320 has a different philosophy: the input stick never moves other than by the hand of the pilot. This main distinguishing feature between the two types of autopilot does not exist in an A320. It does in a B777, which also has an inner loop autopilot plus envelope protection, and an outer loop autopilot which actually does move the column/yoke. So the B777 is functionally identical to the A320 in every meaningful way, plus retains the feature of moving flying controls. It has no mechanical linkage, and it uses two separate sets of actuators for this function: one set for the surfaces, and one for the flying controls.
This is the situation on both Airbus > A320 and Boeing > 777, when all systems are functioning. These aircraft are all dynamically stable: aerodynamics take care of bringing back the aircraft attitude to a neutral position. The F16 for instance is aerodynamically unstable in order to provide high manoeuvrability: it always wants to pitch and roll already, and the flight control computers provide constant rapid inputs to elevators and ailerons to maintain attitude by active control. This is not the case with passenger aircraft, which do not need to be as manoeuvrable as a fighter jet and just need to bring the passengers home safely.
Therefore, both on the Airbuses & Boeings, the aircraft can fly without the Inner Loop autopilots as well, without any problems. The aircraft just lose their envelope protection and now it is possible to command them into potentially dangerous situations such as fully developed stalls. In both types, it is always possible to disconnect the outer loop autopilot. For the inner loop ones:
- The B737 does not have one.
- In the A320, it is not possible to disconnect the inner loop autopilots if all is functioning correctly, except by pulling circuit breakers for resetting the ELACs as this site indicates;
- In the B777 there is a guarded switch that allows disconnection of the envelope protection as @Cpt Reynolds pointed out.
In the latter two types, the system disconnects features by itself if failures are detected, to allow control in degraded mode. Both manufacturers have implemented a direct mode as the lowest mode, where surface deflection is a direct function of stick deflection. There is however no direct mechanical cable connection to the surfaces, even the direct mode is still an electrical input, so technically still Fly By Wire.
Above picture is from
Aircraft Systems Mechanical, electrical, and avionics subsystems integration Third Edition Ian Moir Allan Seabridge, and shows the top level comparison of the implementation of the flight control loops. The redundancy features are different, functionality is much the same: both implementations prevent excessive AoA that brings the aircraft into a stall, and excessive bank angle.
So although the functionality of modern Airbuses and Boeings is very similar, we do often see the myth that Airbus ranks the automated systems higher than the pilot. There is a difference in user interface implementation, in that Airbus has sticks that are not coupled with each other, and have no means to be driven other than by hand. At the time of implementation, this was a new feature that pilots were not accustomed to, as was the envelope protection system. New features can be poorly understood by inexperienced users, and the aviation world is conservative: when the DC3 introduced a 3/4 control wheel, pilots were protesting that they might be gripping the bit of the wheel that was missing and that they were used to.
Ultimately, the most relevant fact is that both Airbus and Boeing have excellent and very similar safety numbers, and that both manufacturers make aircraft that enable the pilots to bring the passengers home safely in 99.9999999 % of flights.