They are both measures of effect size, and what you have described is true of many measures of effect size. Otherwise, they are quite different (for multiple regression). In single regression, they are the same.
The standardized coefficient is a regression coefficient in the units of standard deviation. That is, a for a 1-standard deviation increase in X, we expect Y to increase by B standard deviations. This can be useful to compare the effect of variables if there is some reason to think moving a standard deviation in one variable is equivalent to moving a standard deviation in another (which, so argue, is not often true). This says nothing about how much variance in Y is explained by X; it just quantifies the relationship between X and Y in a unit-free way.
The squared semi-partial correlation is a type of R2 measure that described the proportion of variance in Y that is explained uniquely by X. It says nothing about the magnitude of the relationship between X and Y; that is, it tells us nothing about how Y changes as X changes. You need to have an understanding of explained and unexplained variance to make any sense of this measure; it describes the reduction of randomness in Y due to your modeling choices.