If an NPC is after you repeatedly (for example because of a mission) the only way to shake them off for good (at least until the next jump - they may find you again in the next system, or the next station) is to win the interdiction.
To go silent running and escape to supercruise is a bad idea because if they interdict you again, you don't have your shields back up and your thrusters and FSD is now vulnerable and you might not be able to disengage in time.
By and large, silent running in combat is only worth it when your shields are already down and you need to break the target lock. If turning on silent running means you lose your shields, you're basically just limiting your own tactical options.
And of course SR is fun for smuggling.
But it is under no circumstances some sort of magical invisibility button and is almost always worse than the alternative.