What's the difference? Well, aside from Caliban's assessment (which I feel is correct) there's a number of differences...
1) Time. To raise and train a pack of dogs takes a lot of time that you could not generally spend adventuring. Animal Friendship takes one action per casting. This becomes even more apparent when you're dealing with wild animals, but it is still an issue with domesticated ones.
2)If you cast Animal Friendship, teaching simple tricks does not require a skill check. No matter what your skill modifier, there's always a chance of failure.
3)You asked "What if I raise a pack of dogs who are loyal and loving...?" You'll note that "loyalty" does not appear in the Handle Animal skill - Handle Animal does not at all guarantee the level of loyalty one needs in combat.
This last one is the biggie - while we've all heard stories about a dog who is loyal enough to amazing things for it's owner, the fact of the matter is that such dogs are rare, and you cannot train the animal to have such loyalty. It is a matter of temperament. The Animal Friendship instills this temperament magically. If you went by mundane means, you'd have to go through training on a whole lot of dogs before you found ones suitable to the task.