The info on Harvard at #17 is very good information. Generally speaking, an average Abitur indicates a higher level of academic expertise and exposure than an average US high school diploma. However, there are excellent US high schools, both public and private, that would rival a good Abitur.
Since there is so much variation (public vs. private, one state vs. another, one school district vs. another), colleges (no degrees above BA/BS) and universities (at least one Master's program and often divided into "colleges" or "schools") do put a good amount of stock into the big, national tests (SAT and ACT--some schools accept both, some only one). They also look at a number of other things, including the AP courses mentioned in #1 (these are graded 1-5, I think, with 5 the best; the credit at the college or university is given according to the school's discretion, often for 3-5, 4-5 at better schools, maybe only for 5s at the best), courses taken generally (there's always a minimum core of academic courses for graduation, but then there are also "guidance counselors" who are supposed to help students take the right courses for college study), rank in one's class, the quality of the HS itself, etc., etc., etc.
I have some college students here in the US, where I teach, who would have done quite well in the German system and others who would never have received their Abitur.
I would be careful about generalizations about the US system of higher education. There are just too many types of institutions with all sorts of different emphases to speak of the general importance of rote memorization vs. extensive, intellectual engagement in the US. The best schools emphasize the latter.