The Earth's tilt to the eciliptic is the cause, but distance has nothing to do with it. (During the Northern Winter, the Earth is actually closer to the Sun due to the shape of its orbit.
What happens is that because of the tilt, during winter, the Sun's light strikes the ground at a lower angle and the same amount of heating is spread over a larger area. Also, It means that a smaller percentage of the Winter hemisphere is in sunlight, which shortens the time of daylight. This effect increases the further you get from the Equator.
At the Equator the sun moves from 23° South fo vertical to 23° North of vertical. So the heating and the length of daylight remain fairly constant over the course of a year.
It actually has nothing to do with the distance of each hemisphere from the sun; rather it's the angling and length of day. When your hemisphere is tilted toward the sun, you have a longer daytime and feel the sun's rays more 'head-on.' When you're tilted away, daytime is shorter and you feel the sun's rays more obliquely.
At the equator, days are always 12 hours, and the sun shines from the same angle: it is in the north during northern-hemisphere summer, and in the south during southern-hemisphere summer, but always at the sama angle.