  We show that the distance in total variation between $(\mathrm{Tr}\ U,
\frac{1}{\sqrt{2}}\mathrm{Tr}\ U^2, \cdots, \frac{1}{\sqrt{m}}\mathrm{Tr}\
U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or
symplectic matrix of size $2n$ or $2n+1$, is bounded by
$\Gamma(2\frac{n}{m}+1)^{-\frac{1}{2}}$ times a correction. The correction term
is explicit and holds for all $n\geq m^4$, for $m$ sufficiently large. For
$n\geq m^3$ we obtain the bound $(\frac{n}{m})^{-c\sqrt{\frac{n}{m}}}$ with an
explicit constant $c$. Our method of proof is based on an identity of
Toeplitz+Hankel determinants due to Basor and Ehrhardt, see \cite{BE}, which is
also used to compute the joint moments of the traces.
