The service life of a cruise missile engine is a few hours at most, while airplane engines lasts 1000 hours at least. This alone makes things orders of magnitudes easier.
Also the engine accounts for less than 5% of the weight of a missile but more than 15% on a jet fighter, so less effort could be spent on weight reduction.
Additionally, for surface-launched cruise missiles, the weight budget of the missile is almost unlimited, meaning it could carry arbitrarily large amount fuel to compensate for bad fuel economy. The limiting factor for the range of the missile is more often navigation and homing, rather than fuel and fuel economy.
Finally, a cruise missile flies a at very predictable altitude and air speed and doesn't maneuver much, and always runs at full thrust. As a result, problems like compressor chocking and vibration only needs to be solved for those relatively simple working conditions.
As a matter of fact, other than some long range air-launched cruise missiles that are more constrained by weight and size (e.g. AGM-86), most cruise missiles below 300 km range are quite OK with turbojets even today.
As mentioned in the comments here, those comparisons are only between a cruise missile and military airplane. If you add civilian/commercial (passenger) airplanes to the scene, you'll have to deal with even more complications, such as:
- State of the art fuel economy
- Near-zero failure rate
- Tens of thousands of hours of service life
- Foreign object ingestion
- Arresting engine fragments, had the engine been damaged while in operation
The list goes on and on. In short, an cruise missile engine is a minimum use case, i.e to generate sufficient thrust with acceptable amount of fuel for relatively short period of time, but airplanes engines, especially commercial ones, are concerned with much beyond that.