In general, I feel that the answer depends on the situation you are operating in. As you know, standardized coefficients are expressed on a standardized scale, and capture the ratio of the standard deviations of Y and the X of interest.
On the other hand, the semipartial correlation coefficient is the correlation between the criterion and a predictor that has been residualized with respect to all other predictors in the regression equation. Note that the criterion remains unaltered in the semi partial case. Only the predictor is residualized. Thus, after removing variance that the predictor has in common with other predictors, the semi partial expresses the correlation between the residualized predictor and the unaltered criterion.
What are the advantages of using the semi-partial correlations vs standardized coefficients? In general, the advantage of the semi partial is that the denominator of the coefficient (the total variance of the criterion, Y) remains the same no matter which predictor is being examined. This makes the semipartial very interpret-able. Also, the square of the semi partial can be interpreted as the proportion of the criterion variance associated uniquely with the predictor. This is useful.
Furthermore, you can also use the semipartial to fully deconstruct the variance
components in a regression analysis. How? Each squared semipartial represents the unique variance of that predictor shared with the criterion. Thus, the sum of all squared semipartials is the total unique variance.
In terms of broadening the comment & talking about uses, I note that each partial correlation coefficient is expressed on a different scale. This can make interpretation more difficult. On the other hand, semipartial correlations all on same scale (total variance of Y) do facilitate better comparison across predictors.