They're cut differently. I took a photo of two K10s for you. Both have been washed a few times, so you can see how they end up after shrinking (washed in cold, dried on extra low).
The Anvil (bottom) is wider and a tiny bit shorter (not by much). The main difference is the sleeves and neck hole. The sleeves on the Anvil are much much bigger than the AA. Also, you can't really see it in the photo, but the Anvil neck is much wider too. I used to buy AA K10s for my 7-year-old. They were a little big, but looked fine (and he could grow into them). The Anvil K10 looks silly on him because the sleeves are way too big and the collar falls off of his shoulder when he runs around. I'll probably try an Anvil K8 for him the next time I buy him a shirt. If that's still too big in the arms and neck, I think he's out of luck until he gets older. From the size chart, it looks like the K6 is 4 inches shorter than the K8, so it wont be long enough.
Hope that helps!