As far as I understand, r tells us the strength of the relationship and on a scatterplot, it is indicated by how close the data points are scattered around the regression line, where the closer the points, the stronger the relationship (and higher the |r| value). On the other hand, beta is the standardized B, which is indicated by the slope of the line. How are the two equivalent?
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https://stats.stackexchange.com/questions/32464/how-does-the-correlation-coefficient-differ-from-regression-slope
has the full story
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The simple answer is that the slope of the regression line (which Beta estimates) is also the estimate of the relationship between the X and Y. Thus, the two tests are the same.
I did not intend to suggest that r = beta, but that they are related 100%. See the few pages from a presentation that I have attached. On the last page, you will note that beta = r * s(y)/s(x). The previous slide shows how beta can be derived.
The "standardized beta" of X on Y and "standard deviation" between X and Y is the same thing.
When you have only two variables (X and Y) the standardized slope (beta) is formally equivalent to Pearson's r. If you have more than two variables the relationship no longer holds.
Linear regression is generally more useful than correlation because the unstandardised coefficients are in the original units and more readily interpretable than beta or r (and because it can be readily extended to more complex models with multiple predictors etc.).
Conceptually the correlation is the linear relationship between X and Y capture by the covariance between X and Y and then scaled by the SD of X and Y. It indicates the change in Y assocatiated with a one SD in X. Thus r = 0.5 indicates that increasing X by 1 SD is associated with in an increase in Y of 0.5 SD.
The regression slope b is the linear change in Y associated with a one unit increase in X. If you standardise X and Y (subtract mean and divide by SD) you scale the slope to be in SD units and hence it is equivalent to r.
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