Variance measures the dispersion of an estimator around its mean, whereas the mean square error measures the dispersion around the true value of the parameter being estimated. If the estimator is unbiased then both are identical.
$V(T) = E[T-E(T)]^2$
whereas
$MSE = E(T-\mu)^2=E[T-E(T) +E(T)-\mu]^2 = V(T) +[Bias(T)]^2$
If $T$ is Unbiased for $\mu$ then $E(T) = \mu$, so $[Bias(T)]=0$. Hence $V(T) = MSE(T)$.
Hope you understand this...