How important are standardized test scores compared to other pieces of the application?
Standardized test scores are important because they are a standardized measure that all students can be compared to.
With that said, that is what they are, Know the school you are going to, know their score range, and do your best to do well on standardized tests.
Yet, it is also just one part of the admissions application that helps compare you in a quantitative way to other students. Make sure not to only focus on the SAT or ACT and forget about the other things in life such as student organizations and clubs, sports, and jobs, volunteer work, and all other aspects of life that represent you as an individual and student. Someone who isn't just a number but a complex individual.