I find silent running crucial when trying to dock with stations with a bounty and avoiding getting killed when I've bitten off more than I chew.
Silent running as others have mentioned immediately reduces your external heat signature which makes you very hard to target. As a result, it is difficult to be scanned and difficult to get a gimble lock or any other lockon.
Where people screw up Silent Running is that they don't power down systems afterwards so that they can stay silent.
Combat avoidance:
If you are getting creamed and you can't immediately jump out, you can hit silent running to disappear momentarily from your enemies screens. Then turn off flight assist and boost away in a random direction. With flight assist off, you now can stay going indefinitely in that direction without further thrust. Immediately go to your right CRT and start shutting off all your systems, including thrusters except vitals (like life support). By the time you turn off all your systems, you should be getting close to 100% heat. Now turn off silent running. You will initially bloom on your enemies radar as you dissipate all that stored up heat, but then since you are generating nearly zero heat you will almost immediately disappear again. Yes, you are defenseless at this point, basically a small tumbling piece of metal, but most likely your enemy won't be able to target you and eventually will give up looking or will start looking the wrong way and you'll eventually drift out of his sensor range. After about 3 minutes to 5 minutes, turn on your sensors and see if he's still around...This will make you a little bit more detectable, but not much. If the coast is clear, turn on your systems and frame shift jump the heck out of there.
Busting into a space station.
You've got that cargo full of slaves, that you want to black market, or you just copped some stolen death star plans from some wreckage and you now need to enter the space station to get rid of them. If you go in the typical way, chances are station security will scan you, and either open fire, or jack you with a huge fine that makes those death star plans worthless. So how do you get in undetected. Simple strategy that works every time. Take a wide turn around the station and hover out of the front of the station at about 7.25 clicks. This is too far out for patrols to care about you and is a great place to start your run. Now power down all your systems. If you power enough of them down, you'll see the ice forming on your windshield. Keep life support on, after all, no need to evacuate the cabin for this. The idea is you want to start cold enough that you can get into the station with silent running activated and not do internal damage. Once you are nice and chilly, turn on your sensors and ask for clearance to land. Once you have clearance. enable silent running and turn thrusters on and make a bee line to the mail slot. You are going to raise heat fast so don't dilly dally. You should be able to get through the mailslot without being scanned. Once you are inside the station, turn off silent running and vent all your heat on that poor pedestrian and his poodle making a doodle beneath your ship. Now you can land normally and get rid of those contraband items through the black market. Be sure to turn on all your systems while on the pad.
Some heroes just go in at full speed and try to "beat the scan". However, that can get you killed if someone is coming out, or if you fail to stop before hitting the far wall. I prefer going in silent with full control of my ship. I can ususally vent my ship at 70% heat or less which is plenty of time and I have yet to get scanned.
So Silent Running definitely has it's uses, but it is temporary at best.