The prediction at the average of the covariates is different from the average of the predictions I read in Stata manual : "The prediction at the average of the covariates is different from the average of the predictions" after a logistic regression. 
If I compute predictions for a linear regression (with command Predict that uses own values of covariates for each individual) and I find the mean "a" of my predictors. Then I compute predictions in holding values of my covariates at their means and I find the mean "b" of my predictors. 
Have "a" and "b" the same value?
 A: The answer is no for logistic regression, but yes for linear regression. Why? Because the conditional expectation function in linear regression is linear!
Logistic regression case:
The logistic function $f(x) = \frac{1}{1 + e^{-x}}$ is non-linear. You almost certainly do not have $E[f(x)]$ equal to $f(E[x])$ (except in absolutely degenerate cases like $x = 0$ for all data).
The prediction at the average of the covariates would be: $$f\left(E[\boldsymbol{x}]'\boldsymbol{b}\right)$$
The average of the predictions of the covariates would be:
$$E\left[ f(\boldsymbol{x}'\boldsymbol{b})\right]$$
In any realistic setting, the two would not be equal.
Linear regression:
On the other hand, for linear regression, the conditional expectation function $f(\boldsymbol{x}) = E[y \mid \boldsymbol{x}]$ you're estimating is linear! For a linear function $f$, (eg. $f(\boldsymbol{x}) = \boldsymbol{x}'\boldsymbol{b}$), you have $E[f(\boldsymbol{x})] = f(E[\boldsymbol{x}])$. Intuition: $\boldsymbol{x}'\boldsymbol{b}$ is a sum, taking the expectation is a sum, and you can reorder sums any way you want and it doesn't matter.
For linear regression:
$$E[\boldsymbol{x}'\boldsymbol{b}] = E[\boldsymbol{x}]'\boldsymbol{b}$$
$E[\boldsymbol{x}'\boldsymbol{b}]$ is the average of the predicted values while $E[\boldsymbol{x}]'\boldsymbol{b}$ is the prediction of the average.
Note: I use bold symbols to denote vectors. I use $\boldsymbol{b}$ to denote the vector of estimated coefficients (either from linear or logistic regression, depending on the section).
