[RobotGymnast (OP)]

The Earth is very, very far from the Sun; and yet, the tilt of the Earth's axis somehow makes such a dramatic change in seasons, despite the incredibly low change in distance (in relation to the total distance). Why does that little change in distance make such a big difference?

[Vern]

I don't think it is the change in distance that makes the difference. It is the change in the position of the sun as it passes overhead. In the northern hemisphere, the sun's track through the sky is further south. At the north pole the effect is so pronounced that the sun doesn't rise at all; that makes it cold there. 

[daveshorts]

Basically because it isn't the change in distance that is important, it is the change in angle. In the northern hemisphere summer the earth is oriented so that it is pointing towards the sun, so the sun is higher in the sky
Winter:
[diagram=434_0]
the northern hemisphere is pointing away from the sun so the sun is at a much lower angle and closer to the horizon
Summer:
[diagram=433_0]
In summer you are pointing towards the sun so the sun is higher on the horizon.

The angle changes how spread out the sun is.

If the sun is high in the sky a certain amount of sun is spread out over an area of ground
[diagram=431_0]

If the sun is at a low angle the same amount of light is spread out over a much larger area
[diagram=432_0]
So there is less sulight per square metre and it is colder.

[lyner]

That is the classic  explanation and it, of course, is relevant. But why does no one mention the length of day during winter and summer?
Six hours of sunlight (direct or glancing) will involve a lot less energy than eighteen hours. Whilst both effects contribute to the result, it is surely easier to appreciate that, with the fire turned on for three times as long, a room will get warmer.

[DoctorBeaver]

Does the depth of atmosphere through which the sunlight travels have a noticeable effect? When the sun is overhead it only has the atmosphere's vertical depth to traverse. When the sun is lower in the sky it is at a more acute angle and therefore has more atmosphere through which it must travel.

[RobotGymnast (OP)]

Alright that makes sense... although I have to comment on your use of "whilst"; isn't "whilst" for use just before a word that begins with a vowel?

[yor_on]

Isn't that a dog.
Or to make it more precise, a bow-wow?

A vowel?
Sounds very near a inebriated Swede calling on his 'vovve'

- Come here vowe vovvel, well, to me it does..

[RobotGymnast (OP)]

o_O those are all WAY cooler definitions for "vowel" than mine. I'm going to start using those in English class.

[DoctorBeaver]

isn't "whilst" for use just before a word that begins with a vowel?

No

[lyner]

Does the depth of atmosphere through which the sunlight travels have a noticeable effect? When the sun is overhead it only has the atmosphere's vertical depth to traverse. When the sun is lower in the sky it is at a more acute angle and therefore has more atmosphere through which it must travel.

At a very low angle, there can be hundreds of km of 'thick' atmosphere in the path of the light but that will warm up just the same.

[Damo the Optics Monkey]

Nice explanations above! 

here is  very nice explanatory website that also may help.

Just as an interesting addition to this discussion - This article describes how the Martian axial tilt (25 degrees) also results in seasons on our planetary neighbour.