FBW is a well proven system.
Boeing and Airbus implement it in different ways.
The Airbus implementation will prevent the crew from stalling the aircraft while Boeing assumes that anyone flying a transport class jet is at least minimally capable and will allow the aircraft to perform whatever control inputs the crew make.
Both the B737 and the Airbus single aisle aircraft have performed well in service. One is about as safe as the other.
I fly on both B737s of various generations and sizes as well as Airbus A318/319/320/321 aircraft on a regular basis and have no problem flying on either maker's aircraft.
Their hull loss records are very comparable, and hull losses among first world airlines have become quite rare.
Relax and enjoy your trip. It's not as though you'll be on an uncomfortable regional jet, although the CRJs and ERJs have a pretty solid safety record as well.