There are many ways in which breeding is different from natural selection. First of all, breeding is a lot more controlled, direct, and efficient towards a single goal. Animals or plants are picked out of a group and separated by people, which leads to more rapid changes.
Natural selection is a much slower process.
Some die, some live, but a few more of the less suited organisms die. Sometimes conditions are less harsh and more survive in general, and sometimes conditions change. In every way, natural selection is slower and less efficient than breeding. Also, breeding starts out with a more direct goal. Whereas in nature it does not matter how they changed, as long as they ended up better adapted, in this case better able to keep warm.