Reason for the source of fetal bovine serum
The reason fetal bovine serum (FBS) is used is because it supports a wide variety of cells and really supports their growth.  It has a high concentration of growth factors, but also a low concentration of antibodies.  So it promotes growth in almost any cell or tissue type without also having antibody reactions with it.  

Adult cows blood will have the opposite concentrations, higher levels of antibodies and lower levels of growth factors as they have already grown.

There are currently several research projects looking for a alternative or synthetic FBS, but nothing has worked as a total replacement.  There are a few substitutes for specific purposes, but nothing with the diversity of uses that FBS has.