The old days... turboprops vs turbojet.
Turboprops are constant power engines, from the moment you go full power, you get full power. However, once the speed builds up, it looses efficiency because of almost sonic airflow of the air passing the prop. You need to get the prop RPM down. Excellent for short runways. Not so excellent for high speeds.
Turbojet engines (single axis) are constant thrust engines, where power depends on the speed you are travelling at. So they initially do not have the same performance as a turboprop. Longer runways needed, usually higher speeds as well because of no propwash. But excellent for high speeds. They keep generating thrust when coming closer to the speed of sound.
Both are based on the same "core" which is very efficient when working at high rpm, high pressures, high temperatures. With a prop, no problem at low speeds. Use a gearbox to get the rpm down. But the more power you need, remember that the gearbox has to "transport" all this power.
This gearbox, even with an efficiency of 0,01%, will generate huge amounts of heat. So hot it would immediatly contaminate every drop of oil in the gearbox and lead to immediate failure of the unit. So... not very good for propelling a 777 (even if it would have a max speed of 250kts).
It's hard to "upgrade" the turboprop above a certain power level.
However, turbojet engines went high-bypass. Efficiency in the lower thrust ranges increased because of very sofisticated engineering (turbine blade technology etc...) that, considering the possibility of higher speeds, the lower noise level, increase of performance at low speeds... a small turbofan became a better option compared to a normal turboprop. Turbofans can (after some evolutions in engineering) now be "downgraded" in the section of turboprops.
Development of the gearbox has not halted however, P&W are working on a "geared turbofan", which is in the end the ultimate mariage of turbojet and turboprop. However, you won't see anything like it on a 777 (yet?). A318 is the target for this engine I believe.
Most airlines sticking with the turboprop do this because of: really small airstrip, history, passenger amount. There is no jet version of the Beech 1900 (yet). The Dash 8, well, who is buying it? I wouldn't say the aircraft is still very popular... on the contrary...
Just look at the world of the private aircraft, the "real" private jet has arrived...