In 1 and 2, it allows you to run and sneak at the same time. In 3 and NV, running no longer factors into a successful sneak attack. In 3, it also adds 10 points to your Sneak skill.

Accorind to my notes for builds from Fallout 1 to New Vegas:

-No builds in Fallout 1 and 2 take it. I tend to spend my level 6 perk slot on Bonus Move, Bonus Ranged Damage, or Bonus Hth Attacks.
-In Fallout 3, only the stealth build takes it. Not the Critical build that or both non-critical builds.
-In New Vegas, all builds that use criticals take it.

I have questions about the perk for all the games:

-Is it worth it in 1 and 2?
-If you attack primarily out of stealth in 3, is it worth taking?
-In New Vegas, is it good for both ranged or melee, or is it more useful for one and not the other?

Can't speak about 3/NV, but as far as 1+2 is concerned, Silent Running is only useful for speedruns. Running while stealthy helps cut some realtime.