There's no such thing as "killing better." Dead is dead.
More powerful calibers do more damage, cause more bleeding and destruction and will cause that death to come more quickly.
The weaker of two rounds will simply cause less destruction. If the hit is in the heart (for example) it makes no difference, but if the weakness of the weaker round means it DOES NOT PENETRATE TO THE HEART......that DOES make a difference.
Yes, more powerful rounds do MORE damage. It makes a difference on marginal hits or where greater penetration was needed or where heavy bone was hit and the weaker round blew up on bone and failed to penetrate.