On a pilot-vs-FBW system perspective, as mentioned in the comments to the question, the most notable difference is the authority priority: Boeing trusts more the pilots:
they can supersede the Fly-by-wire automatic commands by exerting enough force on the controls Airbus trusts more the FBW system: if the system is fully functional ("Normal law") it will always have priority over the pilots to protect the aircraft and keep it in the envelope, but they still have room for "non-standard" maneuvers, e.g. commanding bank angles between 33° and 60°.
The autopilot can be activated and de-activated at will by the pilots, and it sits on top of the FBW envelope protection.
One of the most famous examples where this comparison emerges (and the subsequent debate about which approach is better) is the China Airlines 006 accident.
I won't go into detail, there is the accident report that will provide much more information that I can remember, but suffice to say that this 747 was recovered from a quite unusual attitude As mentioned in the wiki article, the report states that the passengers were subject up to 5 gs during the maneuver and the stress on the horizontal tail surface has been so great that as a result it was damaged: The usual argumentation goes along the lines that, would this had been an Airbus, the pilots would not have been able to save the aircraft.
See for example this PhD thesis at page 184 of the pdf, section 6.2.3 "Pilot Authority at the Boundary":
It is likely that if the aircraft had prevented the crew from initiating control commands that would lead to aircraft damage, the aircraft (and passengers) would have been lost.
The problem with such a comparison?
It focuses on the recovery after the unusual attitude was achieved and totally forgets about the sequence of events that lead to it or, to the very least, assumes that an Airbus would have responded in the same way to the preceding events (then why make a comparison at all?)
An honest comparison would take into account this and acknowledge that an Airbus in Normal Law would not enter a spiral dive in the first place, even with an engine out, as bank protection/attitude-hold would prevent it.
aviation.stackexchange.com/users/1467/Federico
Boeing aircraft primarily feature a traditional "yoke" system, which allows pilots to directly control the plane:
In contrast, Airbus aircraft use a "fly-by-wire" system via a joystick which transmits electrical signals to control the plane: Airbus aircraft limit pilots' capabilities in situations that require extreme action to be taken; the computer may prevent the pilot from pushing the plane past its safe ranges, which could be necessary in case of an emergency.
Boeing aircraft leave ultimate control mostly to the pilot.
aviation.stackexchange.com/users/97/Danny Beckett
Big difference is on Boeing with a yoke, it is like two linked steering wheels in a car.
The pilot not flying can easily see what the other pilot is doing with his controls, as the yoke in front of him moves along with the other pilot's inputs.
On the airbus models with a side stick, the pilot not flying cannot see what the other pilot is asking of the controls.
This contributed to the situation where Air France stalled and fell into the sea without the pilot in the left seat realizing that the pilot in the right seat had his controls calling for full pitch up.
(The opposite of what would get them out of a stall.)
aviation.stackexchange.com/users/6984/retired int'l captain
Airbus (A320+; A300 and A310 have traditional controls) uses flight control laws .
In flight, the side-stick input does not indicate desired position of control surfaces, but desired wing loading and roll rate.
The flight computer takes care of trimming the aircraft for straight flight at current speed and balance.
Boeing uses traditional controls, where the position of control column corresponds to position of control surfaces and force on the control column corresponds to force on the control surfaces.
This means that the pilot has to adjust the trim manually when not using autopilot.
The Airbus system is more convenient, but it has its disadvantages.
The neutral stability means that the aircraft won't tend to maintain speed by changing pitch, so the pilots have to monitor speed more carefully.
It really needs the flight envelope protections to be considered safe.
Boeing has flight envelope protections since 777, so that's not a difference any more.
I don't think they are adding them to new versions of 737 though.
aviation.stackexchange.com/users/524/Jan Hudec
My 2 cents:
There are a few things mentioned here .
The sidestick and autothrust systems of the Airbus are different than the conventional yoke and autothrottle system on the Boeing.
The Airbus technology and quiet cruise was great; the Boeing's tactile feel and crosswind capabilities were very good.
There is a difference in the length of the fuselage, making taxiing different.
The handling characteristics in flight are very similar.
aviation.stackexchange.com/users/65/Farhan
What are the main differences in flying a Boeing vs an Airbus aircraft?  Ignoring cosmetics, e.g. nose/window shape (see  How can I tell apart an Airbus from a Boeing? )
What are the main differences piloting Boeing vs. Airbus aircraft?
aviation.stackexchange.com
<airbus><boeing>
aviation.stackexchange.com/questions/149
aviation.stackexchange.com/users/97/Danny Beckett
It seems that the Boeing planes trust their pilots with more control over them. Whereas the airbus planes use electrical signals from a joystic to fly the plane, giving less control in an emergency. It seems that the Boeing planes trust their pilots with more control over them. Whereas the airbus planes use electrical signals from a joystic to fly the plane, giving less control in an emergency.
It seems that the Boeing planes trust their pilots with more control over them. Whereas the airbus planes use electrical signals from a joystic to fly the plane, giving less control in an emergency.