So classic regression methods (ridge regression, LASSO) only predict the posterior mean $E[ Y | X ]$, while Gaussian Processes give you the full posterior distribution $Y | X$.
It would be very useful if the posterior distribution had low variance in regions where the posterior mean is actually close to the true value $Y$, and high variance in other regions. But that's not what is happening (is it?). So why is it useful to estimate the posterior distribution?
The reason why the full posterior distribution is useful is because is contains all of the relevant statistical information that you can derive given the model.
Given the posterior, you can answer any questions like:
given just a point estimate (the posterior mean) doesn't tell you anything about the width of the distribution (although that can often be derived), or, for non-Gaussian models, the shape of the distribution.
This also gives a precise meaning to the idea that the estimates are "accurate" in cases where the posterior distribution has low variance: you can answer the question: "what is the probability is within some range $\pm \delta$ of a point estimate?"
