Silent Running/Fallout: New Vegas
With this Perk, you now have the ability to move quickly and still remain quiet. You can Sneak, and run at the same time. Without this Perk, you would automatically stop Sneaking if you ran.”— Fallout and Fallout 2 description
Fallout, Fallout 2 and Fallout Tactics
Running doesn't deactivate Sneak. Pure convenience.
Fallout 3 and Fallout: New Vegas
+10 Sneak and moving while crouched doesn't decrease the effectiveness of sneaking (otherwise, the penalty for moving slowly is 10% and 20% for running). The perk drastically increases the effectiveness of stealth and the chance of getting Sneak Attack Criticals on everything in sight. Silent Running completely negates ALL movement penalties, regardless of movement speed.
Detection is measured based on a scale of -100 to 0. At -100, you are undetectable.
Despite being labeled as "silent" running, you still generate the same amount of noise from your footsteps as usual, but non-player characters don't react to it.
J.E. Sawyer's Fallout Role-Playing Game
With this perk, you now have the ability to sneak while running without penalty. Without this perk, sneak chances are halved when running.