I've seen this movie many times, and I've got to say, I didn't see anything like a smile. He just looked to me like he was concentrating. I've always liked Andrews' characterization in this movieso calm under pressure, so intelligent and with such good judgment. A good role model. I hope I would be that composed in a crisis.
That said, though, I don't think he really was that composed. I was just reading the other day about an engineer who saw him while he was inspecting the damage and asked him how bad it was. With a worried look on his face, Andrews brushed by him, saying "My God, it's serious." From what I've heard, both Andrews and Captain Smith looked and sounded very worried, at least in that first half hour after the collision. (After that, they poured on the charm and the good humor in order to prevent panic and yet coax people into the boats.)
Still, Andrews must have been a remarkable person. He was wealthy but not at all snooty. He was a guy who went out on the floor with the men, walking up and down and asking if everything was all right, what could be done to make things better, and when he traveled on the maiden voyages of his ships, many of the modifications he noted were things to make the crews more comfortable and make things easier on them. I've never heard anyone say anything negative about him.

I've delivered the most horrible news to people with an absurd, incongruous smile. It's a normal reaction, which I've also seen in others.