First, make SURE that you will in fact graduate from an American high school. I have met several international exchange students who spent their final year of high school in the U.S. but were NOT given a diploma. The reasoning was that these students did not meet the local/state graduation high school graduation requirements - they were only one year exchange students. If that is the case, you may have to return to Romania and get your diploma there. That would delay your college application process – so clearly, knowing IN ADVANCE whether or not you will be given a U.S. high school diploma despite studying only 1 year in the U.S. this is a very, very important factor.
Applying to American colleges while studying in an American high school could make it a bit easier for you – you can get a lot more help with the application process in the US than in Romania, for example. On the other hand, you will still need your Romanian high school transcript and probably a recommendation letter from Romania. And that can be harder to arrange across the ocean.
Even with an American high school diploma – IF you get it – you will be still applying as an international student. So, no, no extra advantage either for admissions or financial aid.