Also, four engine planes most certainly still exist for a very good reason: long-haul flights. They were more common early on due to reliability concerns, and of course modern engines are far more reliable. That said, there are many advantages to a four engine passenger plane: if a very large jumbo, max takeoff weight might need at least 3. And a four engine flight ceiling can either be absolutely required or at least far far more ideal on long-haul flights, thanks to greater fuel efficiency and better flights as they are well above the weather. Dropping to two engines could risk a fuel shortage on very long and heavy flights.
There are almost no advantages to having 4 engines now. 4 engines were mostly for reliability requirements 25+ years ago. Today, engines are reliable enough in general that there is no overall benefit for the higher reliability rate of 4 engines.
Why 2 is better than 4:
- Purchase cost (2 large engines usually cheaper than 4 small engines)
- Maintenance cost
- Lower weight for the same thrust (e.g., 4 engine has 2x the systems parts as 2 engine for the same level of thrust).
- Less drag (For several reasons reasons. 2 engines have less surface area. Also, the farther outboard you place an engine, the more drag it creates on the wing.)
- Lower climb gradient required if engine fails (FAR 25.121)
Look at the A340 (4 engine) and the 777 (2 engine). They were designed and built at the same time. The 777 absolutely destroyed the A340 in sales, and its being used as a platform for the upcoming 777X. Speaking of which, the largest 777X will have about the same number of seats as the 4 engine 747-8, but significantly better fuel efficiency, despite the wing/engine designs only being a few years apart.
The only benefit for 4 engines is that they're easier to fit under the wing - that's why the A380 has 4. The 747-8 obviously has 4 because the older 747s did. Both those aircraft are selling poorly.