The version received on the Lone Wanderer's 10th birthday is different than the standard BB gun. It has a higher item health (1,000), a much faster rate of fire, no chance for a Critical Hit, and does not degrade with use. It is also a quest item, meaning that it cannot be dropped. There is no way to obtain this version of the weapon outside of Growing Up Fast, except through the use of console commands.
The following is based on Fallout 3 cut content and has not been confirmed by canon sources.
During the escape from Vault 101, the gun and 50 BB's can be collected from the Lone Wanderer's desk. They can offer it to Butch DeLoria to fight off the radroaches attacking his mother (although if his mother dies, he will no longer initiate conversation with the Lone Wanderer and the BB gun will be unretrievable without killing him.)
The BB gun is a rare weapon to come across in the Capital Wasteland. With the Point Lookout add-on they are more commonly found, used by swampfolk.
During "Escape!", attempting to leave the Vault and not picking up everything on the desk, will prompt a message that says "You should get your stuff from your desk before you leave" This includes a BB gun. It is classified as being one of the things that must be taken before leaving. This BB gun is a standard version and does not include the special features of the "birthday" BB gun.