There's so many considerations to killing an animal that it's impossible to simplify to one thing. That said, if I was forced to, the #1 thing you want a bullet

There's so many considerations to killing an animal that it's impossible to simplify to one thing. That said, if I was forced to, the #1 thing you want a bullet to do is transfer as much energy to the animal as possible. All else being equal, big bullets carry more energy. Therefore, they transfer lots of energy. Yes, some smaller rounds expand upon impact and tramsfer lots of energy, more efficiently as well, but the problem is that they may expand in fat or bone, before hitting vital organs. So they might not work as well. In a bear charging scenario, you aren't precisely aiming. You are praying your bullets make it past the extremely dense bones and wipe out vital organs. The best way to do this is big bullet travelimg fast. P.S. - when people say stopping power, they usually mean stopping the animal's heart. This is basically the ability to transfer energy to vital organs, which is a combination of penetration and energy transfer.