What are the main differences piloting Boeing vs. Airbus aircraft?
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.