There's a nice old Neural Computation paper on the relationship between the Laplace approximation and variational inference with a Gaussian proxy posterior:

http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf

In fine, the variational approximation is equivalent to requiring the Laplace approximation to hold on average, where the average is taken under the proxy posterior, as opposed to just "locally." Thus, the mean of the proxy posterior under a Laplace approximation is the point (assuming there's only one) where the gradient of the true posterior is zero; whereas the mean of the proxy posterior under the variational Gaussian approximation is the point that renders the average of the gradient of the true posterior zero. Similarly for the covariance matrix.

There's a nice old Neural Computation paper on the relationship between the Laplace approximation and variational inference with a Gaussian proxy posterior:

http://www0.cs.ucl.ac.uk/staff/c.archambeau/publ/neco_mo09_web.pdf

In fine, the variational approximation is equivalent to requiring the Laplace approximation to hold on average, where the average is taken under the proxy posterior, as opposed to just "locally." Thus, the mean of the proxy posterior under a Laplace approximation is the point (assuming there's only one) where the gradient of the true log-posterior is zero; whereas the mean of the proxy posterior under the variational Gaussian approximation is the point that renders the average of the gradient of the true log-posterior zero. Similarly for the covariance matrix.