I don't get why game companies do this. Same game, but the one on the left is JP boxart, while the one on the right is EU.

There's a number of reasons this can happen, a few just off the top of my head would be;

The box/cover art is created by the publisher, the developer usually doesn't have a say unless they also publish it. You can see on both of your examples that the games have different publishers for the different regions, 505 vs Arc and Spike vs Ubisoft.

The American/European publishers know that the Japanese cover art won't sell as well in their region as it would in Japan and vice versa.

The name of the game sometimes changes with region such as the game 'Imagine Ballet Dancer' this is fine in the English speaking territories, that doesn't translate properly over to Japanese though so it becomes 'Princess Ballerina', the publisher feels the name now has completely different connotations and they start making radical changes, not just to the box/cover art but sometimes the entire game.

Oh god, the EU one looks like that chick that 4chan tears to shreds on the regular for her weird "art" posts.