The new PS3 is not backward compatible with PS2 games.

All Xbox360 are backward compatible with a majority of Xbox1 games (478 total).

For the list of all compatible Xbox1 games, look here:

Original Xbox Games Playable on Xbox 360

It saves money to not support PS2 games. The PS2 graphical chip is present in all the PS3s which run PS2 games. By eliminating it in later models that chip isn't required anymore, and maybe it allowed them to reduce the size of the motherboard too.

Sony's biggest problem this generation was coming in to cheap. They should have sold the console for $999 day one. This would have allowed them to sell the console to more main stream consumers buy showing what the console was cabable of beyond gaming. At that price it would have been so expensive people would have stopped and wondered why it was that price. Instead Sony wanted to price the console at an affordable price, but that was just not possible. So what they eneded up with was a console that was loosing tons of money, had the gamers all pissed off, and consumers saying I wont pay that much for a gaming machine.

Had Sony launched at a price in which they could be profitable many things would be looking better for the PS3 right now. First and foremost they could have included the missing 512MB of RAM to make the machine undoubtably the most impressive console on the market. Second they could have had a much better and focused marketing champain, because they were trying to sell consoles, and not just enough to be stable while not to many to loose to much money. Third games like Heavanly Sword, and Lair which showed a lot of promise but faile to deliver due to time and money could have turned out much better.

It all boils down to 2 big mistakes by Sony. One, they made an amazing piece of hardware, but stopped just short of making it a leap over other products on the market. Two, they tryed to sell this product at a price which was bad for everyone.