Primarily, the vault number is just a means to identify a specific vault, so that players can easily identify a given vault between other options.

If you consider Fallout Shelter canon, each vault number coincides with an [actual vault][1], in the Fallout universe. In the canon, each vault played host to a specific experiment. Experiments varied, from having a vault with deliberately-dim lights, to having a vault that contaminated its residents with excess radiation, to having a vault comprised entirely of children.

Players wishing to engage in a deeper level of immersion may wish to select a number associated with a known vault, and recreate the experiment. However, there is no game play mechanic that makes obvious game changes, based off the designated vault number.


  [1]: http://fallout.wikia.com/wiki/Vault