What exactly are some fundamental differences between probit model and logistic regression

Question

It seems that both these refer to cases where the regressed (dependent) variable can only take certain values, as opposed to a linear regression.

So what is the difference between probit and logistic regression?

Answer 1

I think both will produce very similar results, however, the difference is the assumption about the distribution of the errors $\epsilon$ in $y=w^Tx + \epsilon$ (standard normal distribution in the probit model, standard logit distribution in the logistic model). The non-thresholded output of the probit model will be the z-scores of a standard normal distribution, whereas the output of a logistic model can be interpreted as probabilities.