Not a very specifc question. Do you mean difference in organisation? market place/share? products? ethos/culture? finance? politcal structure?
From a flight crew perspective, both produce highly automated flight decks, but boeing put the stick in the middle (where it should be) and airbus to the side. It would appear as well, that boeing put more trust in the pilot to do the right thing at the right time within reason, whereas airbus a/craft are 'trained' to bite if crew step out of line.
Also, boeing produce much more attractive a/craft - B787 v A380? No contest.