Packing Rate for passes is defined as "the total number of opponents outplayed by a pass", whilst Contribution Ratings measure the impact of a pass on the scoreline. Therefore, the main difference between the packing rate and Contribution Ratings is that our metrics capture the context in which an action occurs, whereas the packing rate only counts the number of players that were passed by.
Picture this: a right back passing the ball to a right winger could outplay an entire opponent's midfield, whilst the ball is still in possession on the own half. This situation might get a similar or even higher packing rate compared to a number 10 who chips the ball behind the defensive line and thereby provides his striker with a one-on-one opportunity with the goalkeeper.
Our Contribution Ratings measure the actual impact of an action, so we measure how often a pass would lead to a goal in the near future and would thus rate a pass that ends with a big opportunity (one-on-one with the goalkeeper) to score higher than the own-half pass from the wing back to the winger. Besides that, our Contribution Ratings value every action type (such as passes, shots, crosses and dribbles) whereas the packing rate is only relevant for a small set of actions (passes and dribbles).