How come cockroaches can survive a nuclear war but they are killed by insecticide?

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Short answer: because insecticides are powerful drugs specifically tailored to affect their neurological system when directly aimed, and many will leave long-lasting residual effect. Nuclear bombs not directly aimed at cockroaches may miss them underground for long enough to allow for radiation to dissipate enough for their survival.

Long answer:

There’s obviously considerable exaggeration on the widespread belief that cockroaches would survive a nuclear explosion.

Of course any exposed cockroach wouldn’t survive being hit by a missile, nor the massive forthcoming shock wave, not even the sky-high radiation levels. What is true is that insects are generally more resistant to radiation than vertebrates because of their smaller size and filtering exoskeleton, and that some pest cockroaches are well-known for being able to survive on limited nutrition and reproduce astoundingly quickly for their size.

This way, many researchers believe that cockroaches would likely survive for longer than vertebrates within any cities hit by a major nuclear accident or attack. Whether this is true or not, time will tell.

Now, insecticides are chemicals carefully selected to affect the nervous system of insects, causing death as soon as possible while last for long on surfaces (residual effect). They are designed to kill cockroaches. While nuclear attacks are designed to vanquish cities. The right weapon for the right enemy, that’s it.