I think it was probably negligence. But, you actually can make a case for not calling a time out there. Arizona was out of time outs. They needed at least 9 yards on that play in order to have a realistic shot at a field goal on 4th down, and if they didn't get those yards they were facing a game-deciding fourth and very long. It was a very big play in the game and a poor offensive play swings the win probability well over to the Patriots' side.
Maybe the same thing as on the Butler interception was going on. Belichick decided in the game ending craziness, teams don't often have a third and 23 play easily at their disposal and forcing them to run a play within the 40 seconds allowed and without a time out perhaps leads to a rushed play, a personnel matchup problem, or a false start or something. It was a calculated risk, but when Belichick saw that they did get organized and were in position to run a play, they called the time out.
If Belichick hadn't called the time out, the Cardinals had run to the line and ran a hasty play, and Palmer threw an incomplete pass, bringing up what would have been a game deciding 4th and 23 out of field goal range, it would have been hailed as genius. Maybe that's what he was going for. But it just didn't work out. If that's what was going through his head, that would be coaching and a judgment call, not a mistake. As I said, my instinct is that isn't what was going on. I think what happened was that things were moving fast and the coaching staff couldn't run all the numbers quickly enough after the Cardinals ran a surprising play on second and long, and got caught in the switches of being between wanting them to waste time and wanting to preserve time. But with Belichick, you're never sure.