For Berkeley specifically, EECS is one of the most selective majors, due to its popularity. Applied math is in the College of Letters and Science, which does not admit by major.
Once enrolled, it is very difficult to switch into EECS later, because it is filled to capacity, and there is very little space opened up through attrition. In contrast, applied math is not a restricted major and just requires C grades in the prerequisites (though the department recommends that only students who earn B- or higher grades in the prerequisites declare the applied math or pure math major).
Some other schools do it differently. For example, UIUC and Washington allow applying to majors, but may admit as undeclared or to a second choice major students who are rejected from the first choice major but admissible to the school or second choice major. But then entering the first choice major later involves a competitive admission process based on college GPA and possibly essays. Schools like Texas A&M, Purdue, Minnesota, and Virginia Tech admit engineering applicants to a first year pre-engineering program; students must then compete by GPA and sometimes essays to get into their desired majors.