One of the two transactions will arrive before the other at each node. Each node will reject the second one to arrive.
Eventually one of the transactions (you can't predict which I guess) will arrive at a miner and be included in the blockchain. Once that happens the other will be ignored by nodes.
Very good answer.
Here is one where the answer isn't quite as obvious.
Same exact question, except instead of sending 6 BTC, the two people both send 4 BTC?
(I'm aware of the answer to this question. I'm just posting it to provoke some thought in people who might think they know what will happen. The true experts will get it right, but many people who think they know the answer will be surprised when they read the actual answer)