For PvE situations, it helps you avoid scans. It's like being at 0% heat. You cannot be selected as a target or scanned by another ship unless they are very close to you. It should be noted that once you engage a Human NPC, silent running will not work on them. They don't play by the same rules as human players, and can't be fooled by silent running once they are hostile. Being cold/silent does work on Thargoids oddly enough. As long as you aren't directly in front of them, they will have trouble hitting you.
If you're smuggling or something, I'd say you're better off just using heat sinks than silent running. Dump a sink and don't waste any time getting to the pad, you should be cold plenty long, and you'll still have shields in case you bump into some things.
In PvP it's commonly used as a last-ditch tactic once your shields have gone, and they aren't likely coming back before you die. While silent, players will not be able to easily target you unless they are very close, or you are actively shooting at them. Some builds are based around silent running, and use no shields at all, but they are generally at a severe disadvantage against experienced players in shield-tanked ships. There are a lot of counters for silent running builds.
-Emissive weapon effects
-Corrosive weapon effects
-Hitscan weapons (Rails/Lasers)
-NPC flown fighters and NPC security forces
-Missiles
-Rams
The rams are a big one. With a shield, you can put four pips into SYS and make your shield effectively 2.5% stronger (I think it's closer to 2.75, but close enough). You cannot put pips into hull. Without a shield, your modules are also exposed at all times.
You could also argue that the addition of night vision made it harder on stealth builds, but it was never too hard to pick out ships to begin with. Ship lights, thruster emissions and trails have always been quite visible, even against the dark backdrop of space. Still, it's just one more thing for the pile of counters to stealth. It's still a fun way to fly, but man, you'll really be up against it if you're fighting anyone halfway decent.
You can still boost and remain untargetable while in silent running, but you will generate more heat. Not as much as you would expect from boosting, but it is still noticeable. Boosting at a target may put you in their sensor range, but boosting alone isn't going to make you any more visible. You can be at 3000% heat while in silent running, but to an outside observer, you're still an unresolved contact on sensors.