Mark,
The piston engine and the turbine engine both burn fuel. That's about where the similarities end.
In a piston engine, the more fuel you add, the cooler the engine runs.
In a turbine engine, the more fuel you add, the hotter the engine runs.
The piston engine can burn as much fuel as it can grab air - or to put it another way, there's no oxygen left in the exhaust. Power output is totally dependant on mass airflow.
The oxygen ratio in the exhaust is determined by the mixture setting, and power output is depenne engines tend to be dent on a number of factors including mass airflow, induction air pressure (and temperature), fuel/air mixture ratio, engine RPM, engine displacement, timing, etc. It's entirely possible to have substantial unburned fuel in the engine exhaust, or the opposite.
By my understanding, the gas turbine is limited by temperature - you can't put in enough fuel to burn all the oxygen without melting the turbine section. So there's a bunch of oxygen left over in the exhaust (that's what afterburning/reheat uses). Therefore mass airflow doesn't limit your ability to release power from fuel - temperature rise does.
From a practical point of view, temperature is often not the limiting factor in the turbine engine. Various other parameters limit it first, such as torque (turboprop), EPR, or speed (N1, etc). Turbine engines tend to be limited by those factors at lower altitudes, and temperature limited at higher altitudes.
The majority of airflow through a turbine engine isn't used for the combustion process. Particularly so with a high bypass turbofan engine. Through the engine core, however, the part where the burning is taking place, most of the airflow is used to prevent flame contact with the burner walls, and only a percentage of the total mass airflow is used for combustion. Only when combustion temperature reaches a certain temperature does it become limiting, and other factors may well limit the function of the engine prior to that time.
Particularly at low operating speeds, excess fuel can produce excess temperatures and excess inernal pressure, causing compressor stalls and damage to turbine inlet guide vanes and blades. This isn't a problem in a piston engine, where thermal damage usually only occurs at high power settings and with mixtures close to the ideal (peak) setting.