Similarities and differences between correlation and regression If I want to investigate how two continuous variables are linked, what is the difference between calculating the correlation coefficient (Pearson's $r$) versus calculating the (simple linear) regression coefficient?
I see people who, if the regression coefficient is significantly different from zero, talk about the two variables as if they are correlated, which is confusing as it suggests that the two coefficients (correlation, regression) are the same thing.
Having said that, isn't $r$ a measure of the (regression line) slope anyway? I'm confused!!
 A: 
If I want to investigate how two continuous variables are linked, what is the difference between calculating the correlation coefficient (Pearson's r) versus calculating the (simple linear) regression coefficient?

The regression line is $E(Y|X=x)$. Correlation is a quite different object. 
A regression slope is in units of Y/units of X, while a correlation is unitless.

I see people who, if the regression coefficient is significantly different from zero, talk about the two variables as if they are correlated, which is confusing as it suggests that the two coefficients (correlation, regression) are the same thing.

No, only that they are related, which they are. (Their p-values are effectively the same)

Having said that, isn't r a measure of the (regression line) slope anyway? 

Not of slope, no, as mentioned above. If I change from measuring in meters to measuring in mm, my slope changes by a factor of a million, but my correlation doesn't change at all. But they're related.
A: On the intuitive side, I have been thinking about the following. 
The Pearson correlation is a 2-dimensional linear approximation, while the linear regression is n-dimensional linear approximation. Therefore, the latter offers an estimate of the correlation that accounts for a lot of other features that might in/deflate the estimate obtained with the Pearson correlation. 
See this example1, for the Pearson correlation. Consider a map without info on altitude on it and suppose you can move on it linearly (presence of rivers or cliffs do not matter). You know the time you left point A and reached B, then you compute the speed. 
See this example2, for the linear regression. If instead you move on a map with info on altitude and you have to accounts for all a lot of other info on the ground you are facing (i.e., rivers or cliffs), but still the time you left point A and reached B is as in example 1, the value of the speed you will get will be different (very likely it will be higher).
Although the linear regression offers only an approximation of the average speed, it is still better than the initial approximation you got with the Pearson correlation.
Do some of you find something wrong in this example? (your answers will be very useful as I normally use this example in class)
In any case, I hope this example helped to understand the difference between the two techniques.
A: OLS regression tells you more than the (linear) correlation coefficient. Also, the latter is one of the things you get from the former. Here's what you get with OLS:


*

*A characterization of a linear trend describing how Y relates to X. This trend includes:
1a. The slope (aka beta, effect, coefficient, etc. depending of discipline) of that line, which tells you how much you estimate Y will change given a 1-unit increase in X.
1b. The Y-intercept, which may or may not be of interest, depending on the substantive nature of one's research questions.

*A characterization of the strength of association... that is, does the line $Y = \beta_{0} + \beta_{X}X$ describe the data really well, or does it only kinda describe the data. In the former case, most of the observed data points lie on or close to the regression line; in the latter case the data points may lie quite a ways off the line. Usually, this is reported as $R^{2}$, which is the same thing as Pearson's $r^{2}$

*One gets predictions of the value of Y given a value of X complete with an estimate of the uncertainty of that prediction.
Pearson's correlation coefficient gives one (2), but gives only the sign of the slope in (1a), and does not give intercepts (1b), or predictions (3).
A: To focus one just one aspect of the question (@Alexis answer analyzes well the greater picture), the sample correlation coefficient between $Y$ and $X$ is
$$r = \frac { \operatorname{\hat Cov}(Y,X)}{\hat \sigma_y\hat \sigma_x}$$
while in a simple regression $Y = \beta_0 + \beta_1X+ u$, the OLS estimator for the slope coefficient is
$$\hat \beta_1 = \frac { \operatorname{\hat Cov}(Y,X)}{\hat \sigma_x^2}$$
Combining, we have the relation
$$\hat \beta_1 = \frac {\hat \sigma_y}{\hat \sigma_x}r$$
Pondering this last one, I believe it will provide useful intuition.
