MSE vs variance of residuals

Can someone help me understand the distinction between the Mean Square Error and the variance of the residuals [Var(ei)] in a linear regression? I thought they were the same thing (i.e.the MSE was the variance of the error terms/residuals?)

I understand the MSE = SSE (or RSS)/(n-2)

but there is also a formula for the variance of the residuals (in the context of diagnostics - standardised residuals):

Var(ei) = MSE(1-hii)