A lot of it comes down to economics and engine technology. Four engines burn a lot more fuel per passenger than two, and with fuel prices high for the foreseeable future, it seems reasonable that airlines would try to squeeze as much as they can out of two-engined aircraft. Engine technology has gotten better so that regulators and airlines are increasingly comfortable with longer and longer ETOPS times, so you no longer need the four engines for safety reasons on extended over-water routes.

As a side note, I think the 747 v 380 is a bit of a misdirection; it really is 787 v 380. A and B designed the two aircraft based on fundamental differences in viewpoint about the future of the market for aircraft. Airbus saw high future demand for VLAs to operate from slot-controlled or congested airports on high density routes, where airlines would need to squeeze in as many passengers per departure as possible. Thus the 380. Boeing thought that there was a market for that type of aircraft, but not as large of one as Airbus saw. Unwilling to concede the VLA market to Airbus, but not believing the market large enough to warrant the expense of a clean-sheet design. Boeing went with the 747-8. Boeing thought that the future was in long, thin routes and went with the 787. Time will tell which one was correct, and there is the potential that they both were to some extent.