Prop driven planes have mixture control yet jets don't. Why is this? Even turbine powered props like the SOCATA/Daher TBM have mixture control.
Spark-ignition (gasoline) piston engines have separate throttle and mixture controls. This is because power of spark-ignition piston engines is regulated by throttling the air intake, into which the fuel is mixed. The correct fuel/air ratio is ensured by metering the air using the carburettor or with a similar sensor for fuel injection, but for various reasons adjustments of the ratio need to be done, so there is a mixture control (that may be automated with electronic control unit).
Other kinds of engines – diesel and turbine – don't meter air. They always induce all the air they can and burn the fuel lean, in excess air. Therefore they only have one thing to control, the fuel valve.
The difference is because spark-ignited engines (Otto cycle) mix the fuel with air in the intake manifold and then ignite it with spark-plugs. And such mixture will only ignite if it is sufficiently close to the stoichiometric ratio, the ratio where all the fuel just exactly reacts with all the available oxygen. If there is too much air or too much fuel, the excess will absorb the heat, preventing the mixture from heating up enough and the flame won't propagate. So the engine must induce the right mixture, and its pressure is restricted to reduce power.
In contrast Diesel (Diesel cycle) and turbine (Brayton cycle) engines induce air at whatever pressure is available and inject the fuel into the compressed air at a point where it will auto-ignite at the point the spray from the nozzle sufficiently mixes with air – because the air is hot enough from the compression alone in Diesels and because the flame is continuous in turbines. So there is no problem with having excess air and therefore the air does not have to be metered. Only the fuel is metered, so there is only one lever.
Older spark-ignition engines use carburettor. This induces the fuel simply due to drop in pressure created by a venturi in the intake manifold. But they have to be adjusted with altitude. Best explanation I found is that density of air decreases with pressure, but density of the fuel does not, so while the fuel bowl feels the correct pressure, it still injects relatively more fuel as the air density decreases. This requires adjusting the mixture as the aircraft climbs, otherwise it would become too rich and the engine would stall.
Injected engines, including those with “pressure carburettors” that are simple kind of indirect injection, handle this correctly, but you still want to use richer mixture for high power as the engine runs cooler with a bit of excess fuel, or a leaner mixture for efficiency, as the engine has less power, but higher efficiency with a little bit of excess air. So injected engines still have mixture control.
Only engines with electronic control (ECU; also called FADEC, full authority digital engine computer) automate this completely (using richer mixture at full power), but that's rare in piston airplane engines with exception of some experimental (home-built) aircraft with adapted automobile engines.
Newer turbines and diesels also usually have FADEC, but that is mainly providing protection from stalling or overheating the engine with careless power manipulation. And older turbines already had FCU, a simpler and not as good electro-mechanical system with the same purpose. But there is still only one variable it controls, not two like in spark-ignition engines.
Pre-mixing air and fuel in a way that requires manual regulation is only a thing in older, mostly carbureted gasoline piston engines. Most uses have replaced this method by fuel injection in late 20th century.
Automated mixture control for carbureted engines — feedback — was attempted, but wasn't very reliable. With fuel injection, mixture control could be easily automated, and most injection engines have an automated system to manage the fuel:air ratio.
Manually controlled engines have persisted in general aviation due to regulation driving up the cost of new designs. New aircraft designs, such as most LSA (Light Sports Aircraft), also tend to drop manual mixture control.
Diesels, turboprops, and jets supply clean air to the engine, and inject the fuel separately. In a jet, your throttle is your mixture control.
In a piston engine, each cycle of the piston draws in a certain volume of air in which a certain volume of fuel is to be burned, adding heat to the gases in the cylinder. In the ideal case, that quantity of air is only sufficient to burn to completion the volume of fuel mixed in with it on each cycle; that is, the fuel-air mixture is ideally chemically stoichiometric (no excess air, no excess fuel).
In practice, deviations from perfect stoichiometry are dialed in to keep the engine happy (excess fuel in a rich mixture to carry off heat and prevent overheating the heads) or to maximize efficiency ("xxx degrees F lean of peak" operation). The pilot is responsible for managing that mixture with a mixture control.
But in a turbojet (for example) running not on the Otto cycle but on the Brayton cycle, there is always excess air flowing through the engine, to which a variable flow rate of fuel is injected and burned in the combustor cans. More fuel flow means more oxygen gets consumed as needed out of the excess air flowing through the engine, adding more heat to the gas and hence more thrust and a higher rotation speed of the engine, but at the end of the combustion process the exhaust always contains unconsumed oxygen because the engine's combustion process is carried out in an excess-oxygen environment. This is why you can get a sudden increase in thrust by dumping more fuel straight into the hot gas downstream of the power turbine disc before ejecting it out the exhaust nozzle i.e., afterburning.
Now if you tried doing that fuel dump into the combustor cans instead to consume all the oxygen available, then the turbine temperature would promptly go through the roof and the blades would melt!
So, there is no mixture control as such operable by the pilot of a turbojet.
Note that if a turbojet ran like a gasoline Otto-cycle engine, there would be almost no unconsumed oxygen contained in the exhaust gas stream and afterburning would be impossible.
See Jan Hudec's answer below for more detail about how all this works.
There is no need for mixture control in a jet. In a FADEC engine, (Full Authority Digital Engine Control) the computer meters the fuel. In a non-FADEC turbine engine the FCU, (Fuel Control Unit) is an electro-mechanical device that meters fuel according to inputs such as throttle position, inlet temperature, burner pressure, etc.
I think you are mistaken about turboprops having mixture control. I haven't flow a turbine TBM, but neither the C-208 or T-34 both with PT6 turbines had a mixture control. The lever you are thinking about is probably the fuel condition lever. It looks similar, but performs a different function.
The question is not that prop planes have mixture controls, but that piston engines usually come with them.
The reason is exceedingly simple. Aircraft engines manage fuel/air mixture using a system that is very unlikely to fail catastrophically: the pilot. There is no reason why a computer couldn't take over the function, except the excessively convoluted certification process (for many good reasons) that makes it cost prohibitive.
There have been single-lever aircraft piston engines out there (the WW2 era Fw190 comes to mind as an early example), but far too often they have not withstood the test of market conditions and time. Today a few production piston aircraft come with the necessary electronics to forgo the mixture control.
The reason for turbine engines automating those controls is that the tolerances for acceptable fuel/air mixtures are much tighter. In a piston engine key components are operating hundreds of degrees below their melting point. By comparison, a turbine engine might only be 50 degrees short of where things start to dramatically change their ductility. While theoretically a human might be able to manage the fuel/air mixture of a turbine engine as a full time job, the probability of human error causing expensive engine damage (or worse) is just too high. Hence the automation.
The reason jets don't have a mixture control option is because there is no need to change mixture ratio. There is no such thing as 'lean-of-peak' on a jet.
Contrary to piston engines that burn fuel in a series of single explosions, jets burn fuel in one continuous flame. The main drive behind that flame is an equally continuous flow of air through the jet, caused by a compressor and/or the inside shape of the jet. That is why that flow has to be there before you can ignite the flame.
Once the flame is burning, it is the expansion of gas in the flame that keeps the air flowing, much in the way a turbo charger in a car is powered by the engines own exhaust pressure. This indirectly links the fuel flow to the air flow in a more or less fixed ratio.
Contrary to piston engines, there is not a whole lot of need to ever play around with that ratio in jets. Moreover, doing so may very easily result in catastrophic engine failure. A flame-out and a compressor stall are a way bigger deal than for instance a frozen up carburetor, not to mention the risks involved with melting fan blades.
In general, Jet engines are a very delicate expression of an in itself simple and very violent principal, whereas piston engines are a fairly robust expression of a comparatively delicate and complicated principal. This is why it took so long to develop a usable jet engine, once it was invented, whereas the piston engine was pretty much operational from day one on. Thus it is possible to optimize the performance of a piston engine by evading from the manufacturers manual, whereas the best way to obtain optimal performance out of a jet, is by sticking to the manual, in use as well as in maintenance.