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 and since it is one of the Lone Wanderer's possessions because of the message stating that the "stuff" must be taken before leaving. It is the BB gun given to the Lone Wanderer on their 10th birthday. However, it does not have the unique features like the one given. The birthday version is taken away despite being a present and replaced with a normal BB gun that is seen as the one that is given on the 10th birthday.