Because people associate puppies with innocence. It's the same thing with children. You'd be more upset seeing a child murdered than an adult. it's just how it works. In war, we know atrocities happen, like accidental deaths in bombings and crossfire; but then you have these asshats, not even in combat at the time, killing a little puppy in a violent and cruel way. Yeah, it's a lot different.
If an innocent dog was caught by a stray bullet, and a camera guy caught it on film, it wouldn't NEARLY be as troubling as a soldier tossing it ruthlessly off a cliff.