Why would someone confess and others proclaim innocence right up through their execution?

Someone would confess either because they have been offered a deal (like a lesser sentence) tied to their confession, or for other (often psychological) reasons, such as to alleviate the burden of guilt, or to provide closure to the surviving victim(s). Some confessions, however, are a result of coercion and manipulation by authorities.
Proclaiming innocence is mostly perceived as an indication of innocence, especially if persistent to the point of seeming unreedeming (like right before and even during execution). Perhaps this is why some guilty people still proclaim innocence to the very end. But there are many innocent people who die still proclaiming their innocence