Suppose G=SL(2,C) and let H be the stabilizer of a line (so a Borel subgroup).  The matrix $$\begin{pmatrix}a&b\\\\c&d\end{pmatrix}$$
in G acts on projective space by
$$ z \mapsto \frac{az+b}{cz+d}$$
The stabilizer of ∞ is the matrices with c=0, a Borel subgroup.  The stabilizer of both ∞ and 0 is the matrices with b=c=0, a maximal torus.  In particular, a two point stabilizer is abelian (the intersection of two Borel subgroups), and a Borel subgroup is non-abelian.  Hence they are not isomorphic.

This is just a connected version of Giuseppe's answer.