In a simple linear regression
$$y=\alpha+\beta x + \varepsilon,$$
if $x$ is binary we can show that the slope coefficient ($\beta$) is equal to the difference in the outcome variable between levels of the predictor, i.e.
$$E(y\mid x) = \alpha+\beta x$$ so $$E(y\mid x=0) = \alpha$$ and $$E(y\mid x=1) = \alpha+\beta$$.
We then have: $$E(y\mid x=1) - E(y\mid x=0) = \beta.$$
However, in a multiple regression this usually is not the case. When we control other variables, some of the variation in $y$ originally explained by $x$ may be partitioned into the new covariate, which changes the estimate on $\beta$ so that the original relationship of $E(y\mid x=1) - E(y\mid x=0) = \beta$ will no longer be true.
I understand this intuitively and can show it with data, but I'm not sure how to express it mathematically with formulas (?).
An illustration using data in R:
set.seed(1) # generate data outcome <- rnorm(n=100, mean = 10, sd = 1) original_x <- rbinom(n=100, 1, prob=.5) new_x <- rnorm(n=100, mean = 10, sd = 10) # Simple Linear Regression coef(glm(outcome~original_x)) # 0.08680985 mean(outcome[original_x==1])-mean(outcome[original_x==0]) # 0.08680985 # Multiple Regression coef(glm(outcome~original_x + new_x)) # 0.1112377