\bgroup is a synonym for {, which is defined in Plain TeX using \let\bgroup={
It interacts with TeX's "digestive system" in hairy ways: {s and \bgroups start the same sort of groups, called simple groups, and each can be terminated with either }s or \egroups, since they are the same. But when the Tex digestive system encounters them, they are of different catcodes, so commands that look ahead, e.g., in LaTeX with \section\bgroup Title}, can break this matching.
\begingroup is different. It is a Tex primitive, and it matches a different sort of group that TeX accounts for separately, called "semi-simple groups" (a Knuth joke, I assume). Thus a \begingroup must be terminated by an \endgroup, not a }, and vice versa for \endgroup.
I generally avoid \bgroup, and use \begingroup, but \bgroup could be useful if you are messing about with a nested token list.
