From The Vault
Silent Running is a Fallout, Fallout 2, Fallout 3 and Fallout Tactics Perk.
[edit] Fallout, Fallout 2 and Fallout Tactics
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.
[edit] Fallout 3
With the Silent Running perk, you gain an additional 10 points to Sneak, and running no longer factors into a successful sneak attempt. Drastically increases your ability to crouch-run through buildings getting free sneak criticals on everything in sight. This perk also eliminates the added noise generated from having heavier items equipped, so you no longer receive a penalty for sneaking in power armor with a Fat Man in your hands, for example. Together with the Chinese Stealth Armor, the player is able to sneak as efficiently as possible. For combat, consider using the Perforator to remain undetected. Despite being labeled as "silent" running, you still generate the same amount of noise from your footsteps as usual, it still works however.
Note:This perk does not mean that MOVEMENT will no longer factor into sneaking, but RUNNING. This means that running and walking slowly while sneaking will have the same effect, not that you can move around all the time and expect to not get detected.