If the solstice occurs at that moment when the earth is at its maximum tilt, then why does the weather extreme begin that day, and last for so long afterwards?
This question has vexed me since I was a child. In other words, "winter" is said to BEGIN at the winter solstice. But if that is the day when the sun's center is at its highest northward point
1. The Earth's tilt remains constant. The North polar axis is constantly pointed at pretty much the same point in the sky. But as the Earth orbits the Sun the direction of tilt relative to the Sun changes. The Sun is at the lowest point to the south on solstice.
2. The Northern hemisphere builds up and stores heat during the summer. That heat cannot disappear instantaneously. It is in the ground the water and the atmosphere. It is all of the shifting around of heat that creates the wind and ocean currents. So the coldest temperatures aren't until after solstice. It is a kind of thermal inertia.
The further the earth gets away from the sun the more extreme the weather, in this case the winter, depending on your geo-position. As for why it last so long,
The Earth's orbit is slightly elliptical and I have read that the Earth is closer to the Sun when it is summer in the southern hemisphere and consequently their summers are warmer than ours. I haven't been there to test it though.
PS - The Earth doesn't exactly point at the same place in the sky. The planet precesses slowly and a complete cycle takes 26,000 years. This is related to the Age of Aquarius business.