Roxanne's key move is the claim that the two ivory markets are entirely independent. Because antique ivory and new ivory have nothing to do with each other, buying antique ivory can't drive poaching. So she concludes that buying antique ivory is fine, and only new ivory is off limits.

Salvador accepts that concerned people shouldn't buy new ivory, but he thinks Roxanne stops too early. His point is that the markets are connected after all. Since there isn't enough antique ivory to go around, people who want antique end up buying new ivory instead. So even a concerned person who buys only antique is using up the scarce antique supply and pushing someone else toward new ivory. His fix is to buy no ivory at all, which loosens the antique supply and brings new-ivory demand down.

The two agree on plenty: elephants are worth protecting, and concerned people shouldn't buy new ivory. The split is over whether the antique and new ivory markets are linked. Roxanne says they're entirely independent, so what happens in the antique market never touches the new market. Salvador's whole argument depends on the opposite, that antique demand spills over into new-ivory demand.