Logistic vs linear regression

Let's say I run a linear regression model with a binary dependent variable. If I ran logistic regression on the same data would the results be comparable or exactly similar? By results I mean both the beta values and the value of dependent variable. If not why? Also what can I say about linear regression being a subset of Logistic regression or vice versa?

• The models for linear regression and binary logistic regression are both special cases of the generalized linear model (GLM): Linear regression with an identity link and Gaussian conditional distribution (with equal variance), logistic regression with a logit link and binomial conditional distribution. Nov 28, 2012 at 13:05
• I discussed the nature of logistic regression in a slightly different context here: difference between logit and probit models. That answer also briefly mentions the connection b/t linear regression and the generalized linear model. It may be helpful to read it. Nov 28, 2012 at 15:30

Neither method is a subset of the other. Suppose you have response variable $Y$ using covariates $X$. In linear regression, you assume that $E[Y|X]$ is linear in the parameters $\beta$, whereas in logistic regression you assume the log-odds,
$$\log\Bigg(\frac{P(Y=1|X)}{P(Y=0|X)}\Bigg)$$
is linear in the parameters $\beta$. For each method you assume two different quantities are linear in $\beta$. The estimates of $\beta$ will likely be different and have different interpretations as well.