Just going in and saying, hey, look what these guys gave us is not likely to work. Colleges have limited aid budgets, and far more kids (all of whom are deserving) competing for those limited funds than they can afford to pay for. Because of that scenario, they take very seriously their obligation to be fair; they don't want to create an appearance of favoritism, where they throw money at people willy-nilly and without any sound basis -- that's the kind of situation that can result in "oh, look, we gave fantastic aid for all these white males, and ignored the women and minorities who asked," which generally doesn't go over well for oh-so-many reasons. So most colleges are going to have a pretty concrete appeals process, with well-defined reasons for giving more. And "school X gave me a better package" probably doesn't qualify -- maybe unless School X is Harvard and your kid is someone the school really, really wants (but if that were the case, they probably would have offered more up front).
OTOH, if you can present some substantive information that would support the change, and during the course of that slip in that of course she'd far prefer to go there but the other school came up with an EFC that was $XX less, it might work. For ex., when I was applying back in the dark ages, I had narrowed it down to two SLACs that were about $5K apart in tuition. The two schools came up with very similar aid packages -- federal loans, some need-based grants -- but the lower-priced one participated in the National Merit Scholarship program and so offered me a $2K scholarship on top of that, meaning the other school would have cost me about $7K more. I went back to the second school to tell them that I basically couldn't afford it on their offered aid, and I mentioned the extra scholarship. That school did not participate in the National Merit program, but they nevertheless suddenly found another $2K in grant money (but they still didn't make up the $5K tuition delta, and I went to the cheaper school anyway).
Similarly, while I was in school, my mom went on sabbatical, and so her income was cut in half, but since the aid figures were based on the prior year's tax returns, those numbers stayed the same (and tuition had gone up by $1K to boot). As a result, their aid award assumed that fully half of my mom's gross pay would be available to cover my college costs (they also assumed my stepdad paid for everything else so her income was basically "free money" - thanks, sexism!). So I appealed that award, presenting them information about the cut in pay and my mom's other financial obligations/arrangements, and they increased the award quite a bit.
Tl;dr: If you want to negotiate more aid, you need to give them an actual financial reason to justify it -- something they overlooked or misinterpreted.