What should I do if the characters on the front and back of the party application are different in thickness?

It doesn’t matter. The thickness of the handwriting on the party application has no fundamental bearing on whether you will be approved to join the party. This is not an exam. The neatness of the paper and the clear and elegant handwriting may leave a good impression on the marking teacher! The people who discuss your joining the party should basically be people who are familiar with you and understand your performance and requirements, not people who appreciate your calligraphy!