In linear regression, we often get multiple R and R squared. What are the differences between them?
Capital $R^2$ (as opposed to $r^2$) should generally be the multiple $R^2$ in a multiple regression model. In bivariate linear regression, there is no multiple $R$, and $R^2=r^2$. So one difference is applicability: "multiple $R$" implies multiple regressors, whereas "$R^2$" doesn't necessarily.
Another simple difference is interpretation. In multiple regression, the multiple $R$ is the coefficient of multiple correlation, whereas its square is the coefficient of determination. $R$ can be interpreted somewhat like a bivariate correlation coefficient, the main difference being that the multiple correlation is between the dependent variable and a linear combination of the predictors, not just any one of them, and not just the average of those bivariate correlations. $R^2$ can be interpreted as the percentage of variance in the dependent variable that can be explained by the predictors; as above, this is also true if there is only one predictor.