Partial regression plots are most commonly used to identify data points with high leverage and influential data points that might not have high leverage.
Partial residual plots are most commonly used to identify the nature of the relationship between Y and Xi (given the effect of the other independent variables in the model).
Note that since the simple correlation between the two sets of residuals plotted is equal to the partial correlation between the response variable and Xi, partial regression plots will show the correct strength of the linear relationship between the response variable and Xi.
This limits its usefulness in determining the need for a transformation (which is the primary purpose of the partial residual plot).
This article incorporates public domain material from the National Institute of Standards and Technology