The standard errors are different because the variance assumptions in the two models are different.
Logistic regression assumes the response has a binomial distribution, while beta regression assumes it has a beta distribution.
The variance functions of the two are different. For the binomial, if you specify the mean (and $n$ is a given) the variance is determined. For the beta there's another free parameter, so it isn't determined by the mean and would presumably be estimated from the data.
This suggests that if you fit a quasibinomial GLM (adding a variance parameter) you might get closer to the same standard errors, but they still won't be the same, since they would weight the observations differently.
What you should actually do:
if your proportions are originally counts divided by some total count, then a binomial GLM would be an appropriate model to consider. (You would need the total counts, though.)
if your proportions are continuous fractions (the proportion of milk that's cream for example), then beta regression is an appropriate model to consider.