The least squares means are, in a sense, weighted averages. You can obtain similar results by using predict.glm with a type="response" option and then summarising over the variable of interest. E.g. predict the umeployment rates for all the data, and then summarise for all Males / Females and calculate the difference.
SAS's ilink option is doing the same thing by inverting the link (logit) function and turning the estimates from log odds back into probabilities - converting the estimates to the scale of the response variable (i.e. probability).
Be careful reporting differences in probability derived this way. As an extreme example, if there was a sub-population with a Male unemployment rate of zero, would you expect the unemployment rate for Females in that same sub-group to to be zero, or -8.9 percent?
We can rationalize this as follows:
Underlying logistic regression is a latent (unobservable) linear regression model:
$$y^* = X\beta + u$$
where $y^*$ is a continuous unobservable variable (and $X$ is the regressor matrix). The error term is assumed, conditional on the regressors, to follow the logistic distribution, $u\mid X\sim \Lambda(0, \frac {\pi^2}{3})$.
We assume that what we observe, i.e. the binary variable $y$, is an Indicator function of the unobservable $y^*$:
$$ y = 1 \;\;\text{if} \;\;y^*>0,\qquad y = 0 \;\;\text{if}\;\; y^*\le 0$$
Then we ask "what is the probability that $y$ will take the value $1$ given the regressors (i.e. we are looking at a conditional probability). This is
$$P(y =1\mid X ) = P(y^*>0\mid X) = P(X\beta + u>0\mid X) = P(u> - X\beta\mid X) \\= 1- \Lambda (-Χ\beta) = \Lambda (X\beta) $$
the last equality due to the symmetry property of the logistic cumulative distribution function.
So we have obtained the basic logistic regression model
$$p=P(y =1 \mid X) = \Lambda (X\beta) = \frac 1 {1+e^{-X\beta}}$$
After that, the other answers give you how we manipulate this expression algebraically to arrive at
$$\log \frac {p}{1 - p} = X\beta $$
It is therefore the initial linear assumption/specification related to the Latent variable $y^*$, that leads to this last relation to hold.
Note that $\log \frac {p}{1 - p}$ is not equal to the latent variable $y^*$ but rather $y^* = \log \frac {p}{1 - p} + u$
Best Answer
I don't know of SAS, so i'll just answer based on the statistics side of the question. About the software you mays ask at the sister site, stackoverflow.
If the link function is different (logistic, probit or Clog-log), than you will get different results. For logistic, use logistic.
About the real differences of these link functions.
Logistic and probit are pretty much the same. To see why they are pretty much the same, remember that in linear regression the link function is the identity. In logistic regression, the link function is the logistic and in the probit, the normal. Formally, you can see this by noting that, in case your dependent variable is binary, you can think of it as following a Bernoulli distribution with a given probability of success. $Y \sim Bernoulli(p_{i})$
$p_{i} = f(\mu)$
$\mu = XB$
Here, thew probabitliy $p_{i}$ is a function of predictor, just like in linear regression. The real difference is the link function. In linear regression, the link function is just the identity, i.e., $f(\mu) = \mu$, so you can just plug-in the linear predictors.In the logistic regression, the link function is the cumulative logistic distribution, given by $1/(1+exp(-x)). In the probit regression, the link function is the (inverse) cumulative Normal distribution function. And in the Clog-log regression, the link function is the complementary log log distribution.
I never used the Cloglog, so i'll abstein of coments about it here.
You can see that Normal and Logist are very similar in this blog post by John Cook, of Endeavour http://www.johndcook.com/blog/2010/05/18/normal-approximation-to-logistic/.
In general I use the logistic because the coefficients are easier to interpret than in a probit regression. In some specific context I use probit (ideal point estimation or when I have to code my own Gibbs Sampler), but I guess they are not relevant to you. So, my advice is, whenever in doubt about probit or logistic, use logistic!