Coach Scott Robertson said, "We definitely know how hard it is now.
"It hurts because we wanted to put our name on the history board like the others have done and make them more proud."
England proved better at taking opportunities than when they played the All Blacks a year earlier at Twickenham.
"It was a tough Test last year, they were moments away. All Test footy feels like that at the moment. But they've taken their opportunities tonight where they probably didn't last time and we did through these small margins."
Captain Scott Barrett said the Grand Slam was a goal they set out to achieve, and England pulled the rug out from under them.
"We'll quickly put our focus towards Wales and finishing strong. It is hugely disappointing to come away with a loss here. We have to bounce [back] and get better. It'll be a bit of a pebble under the beach towels."
Robertson repeated comments from earlier games, saying the All Blacks had created a lot but hadn't taken advantage of it.
"England finished a bit more than us."
He was pleased with the discipline shown in the first half, but they hadn't built on their 12-0 lead.
"We missed a couple of kicks for the corner, and then trying to be brave and put them in there, and keep the points, and the momentum going, didn't come. We missed them; they got a couple of dropped goals. Come out [after halftime], a yellow card in the second half and the game swings so quickly.
"We know what Test football's like, and we needed to finish more."
Robertson said he couldn't put a finger on why the side was struggling in the third quarters of games.
"We've tried a lot at training to educate ourselves as much as we possibly can around what it takes to keep that momentum, keep pressure on teams. That hasn't happened, so it's frustrating.
"We've put so much work into it and it is definitely an area of growth."
Beauden Barrett's two missed penalty touchfinders caused lost momentum for the side.
"They came back and got a couple of droppies out of it that we went to a point behind at halftime. If we came out and scored again and got them to chase, they would probably get out of their kicking structure.
"They won a lot of air battles. We won a few too, which was pleasing, but not enough."
Scott Barrett said, "England were pretty efficient when they got into our half in creating, and coming away with points, and they probably had a few more entries into our half, and George Ford was pretty accurate with his dropped goal and keeping that scoreboard ticking over.
"We had limited opportunities in that second half and they kicked well. The last couple of games we have finished quite well, but tonight we didn't do that."
England's defence was strong, and efforts to turn them around with more variation in kicking were hampered by England's ability to slow down the All Blacks' ball.
"They probably won the contact area and the breakdown and did enough to slow it down so we couldn't quite get a roll on like we would have hoped."
The All Blacks take on Wales at 4.10am NZT on Sunday 23 November. Click HERE to find out where to watch around the world.
