Why did the turbojet replace the piston engine?
Two more reasons the gas turbine supplanted the piston engine for aircraft use:

Power output. Aircraft piston engines have a practical limit on how much power they can put out, before becoming inefficient. This worked out to be around 3000hp. Two of the largest and most powerful piston aircraft engines that were also reliable enough for aircraft use are the Napier Sabre and P&amp;W R4360, at around 3000hp, developed at the end of WW2. Yes, more power has been obtained from piston engines on land based vehicles, but they're either racing engines whose reliability is too low for aircraft use, or they're far too heavy for aircraft use, such as the 20k+ HP diesel engines that power cruise ships. 

Gas turbines don't have those limits, one contributing factor being the lack of a reciprocating action. With a piston engine, each full stroke results in the piston accelerating to top speed, then coming to an abrupt halt and accelerating in the opposite direction... twice. Gas turbines that just spin can be made substantially larger while maintaining the reliability and efficiency that aircraft use demands. 

Also, gas turbines tend to have a very high power to weight ratio, making them ideal for large aircraft use, where weight is very important. 

Equating HP to thrust isn't simple as HP is raw power, while thrust includes altitude, velocity, and propeller/fan efficiency.

A simplified example was published on Aerospaceweb:


  Luckily, we do have access to data from a NASA report that does
  provide all the data we need to illustrate a sample case. The data is
  provided for a Boeing 747-200 cruising at Mach 0.9 at 40,000 ft
  (12,190 m). In this example, the aircraft's engines produce 55,145 lb
  (245,295 N) of thrust, only a quarter of its rated static thrust, to
  cruise at a velocity of 871 ft/s (265 m/s). Using the equations
  provided above, we calculate the power generated by the 747 to be
  87,325 hp (65,100 kW).


Using that very simplified example, a GE90 producing 115,000 pounds of thrust would be putting out the equivalent of around 160,000 horsepower.  

Also, the maintenance requirements on gas turbines are substantially lower, especially for the high output engines. For example, the very large RR Trent series turbofans have a TBO (Time Between Overhaul) of around 15,000 hours. On the large piston engines such as the R4360, the TBO was more like 1500 hours, and the very large piston engines, especially the radial engines, had a prodigious appetite for engine oil. Plus intermediate maintenance on piston engines that gas turbines don't need, like changing spark plugs, that had to be done frequently. The Convair B36, which had six R4360's, required 336 spark plugs. Not something you could do in your driveway in an hour. 

Some of the gas turbine's reliability comes from it's lack of vibration. Large piston engines, with their reciprocating action, tend to vibrate a lot, which reduces the life of engine and auxiliary components, like fuel pumps and spark plug wires. 

Thus, not only did the turbojet, turboshaft, and turbofan make possible aircraft that wouldn't be practical at all with piston engines, such as large airliners flying at 30k+ feet, they also lowered the cost of maintenance and the frequency of maintenance substantially. 

There is one area where the piston engine for aircraft use is still the better solution, and that is when the engine gets small, below around 500hp. Gas turbines don't scale down all that well. The small ones are not fuel efficient, nor does the cost get substantially lower. 

As an example of this, consider one of my pet daydreams - a single seat Mosquito helicopter. Aside from the ultralight version, two versions with larger engines (and requiring a FAA helicopter license) are made, the XE285, with an 85hp snowmobile engine, and the XET with a 90hp gas turbine engine derived from a backup power generator. The piston engine sells for maybe 2k USD and burns around 5gph, while the gas turbine sells for 10k and burns 8.5 gph.