  We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$,
and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in
each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding
union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent
cone. This is a finer version of Lusztig's result that corresponding special
pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The
proof requires studying the case of characteristic 2, where more direct
connections between the three nilpotent cones can be established. We also prove
that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth
in any characteristic.
