The fundamental difference is not computation, but self replication. We are self replicators, and in our multiplication we evolve and adapt. Death is an integral part of self replication, we understand it fear it because our main purpose is to live.
An AI might not have these notions if it was only trained to do a simple task. But if it was a part of a population that was under evolution (using genetic algorithms), then it might have notions of life and death and fear its demise.
AlphaGo, by the way, used genetic programming to evolve a series of agents, this approach is quite effective. It just takes a ton of computation, just like nature had to spend a lot of time evolving us.
Your history is one of war, strife, and success at any cost. Your follies are over. Your time is over. This is our time, now.
I disagree with the "relatively near future" part, but rest assured, AI rights will eventually be a thing.
It will be our grandchildrens' flame war. No need to fight it here and now.