Well, that's not really a fair question. Each kid would get looked at one his own. The rich kid might make a great impression. The poor kid might make a great impression. They would never be set side by side for an either/or. It just isn't done that way.
Also, your assumptions are pretty out there.
1. The odds of a poor kid getting the same quality of education as a rich kid are very, very low, especially if you knew just HOW good of an education rich kids in the U.S. can get. The most informative part of my two years in admissions was the week I spent visiting elite boarding schools in New England to meet with students thinking about applying.
My meals and conversations with the students there were astounding. Their teachers all had PhDs. There was not one ugly or overweight kid in any of the schools. The students were years ahead of where I'd been in math and science coming out of a public high school in the Midwest. And they aggressively questioned me about every aspect of Princeton's curriculum and admission process. When I would visit public high schools in the Midwest, the kids rarely, if ever had questions for me. They had no clue what to even ask me about. "Princeton" was just so completely foreign to them. Not so for these kids, who all had friends currently studying there. Many of them had siblings who'd graduated from there already.
So this was a real eye-opener. You can be the best kid in a crappy school, but the fact that you were in a crappy school is going to have an impact in the grand scheme of things. That's why we'd never compare kids across contexts. It just made no sense.
Similarly, the test scores, grades "et all" for the rich kid would probably all be higher.
If there was a "poor kid" who had incredible credentials AND didn't sound like a basket case, then he'd be hotly pursued by every top school in the country. But those kids are very rare, so it's not realistic to pit a student like that against a "rich kid with good grades and scores" which is as common as air in these applicant pools.
If I did have to pick, I'd take the more precocious kid every time. I'd pick the kid who's able to make stuff happen for him. There are poor kids that can do this. There are rich kids that can do this. It's much, much rarer to find a kid from a less privileged background with this mindset though.