Kimball, if you haven't guessed, points towards the future. He understands line of action and rhythm in a pose. If you single frame Kimball's work, the poses are very strong silhouettes and the rhythms are pleasing.
By contrast, Klein looks to the past; he started animating at the Hearst studio in the teens and his animation is just fussy. Klein does a pretty good job of capturing the acting style of Oliver Hardy, but his work is less graceful than Hardy himself.
Take a look at these two images. Both are anticipation drawings.

Look at the long, graceful curves in the Kimball drawing. If you look at the left side of the Groucho figure, the curve starts with the hand and travels all the way through the coat tails. If you prefer, you can follow it down the leg to the foot. Those kinds of long curves give the figures a unity and a flow and they're evident in all three characters.

The same lack of rhythm in Klein's individual drawings is evident in his animation. It's not enough to draw a rhythmic pose, your path of action has to be rhythmic as well. Just as Klein's lines are fighting each other, the movement of his character's parts fight each other too. The fussiness in the animation is because the poses and actions aren't tied together with rhythm.
Fred Moore didn't work on this cartoon, but Kimball's work couldn't exist without Moore's. Moore used rhythm to unify individual drawings and unify the sequence of drawings into a coherent statement. Kimball got it and Klein didn't, which is why people talk about one and not the other.