The maximum texture size effects the visual details of the textures applied to the objects in the game.

Unlike video resolution, which sharpens and reduces the jagged appearance of the edges of objects in the game, the texture size effects the details within the object itself.

Take note of the differences in the shots to the right using the various texture sizes in the game.

So why not default to 2048 texture sizes and not be worried about any other format?

First, some video cards simply do not support texture sizes this large. Secondly, the amount of video memory required to deal with 2048x2048 textures is high. Each 2048 texture takes 16MB of video RAM. Think about that for a minute.

The performance ramifications are significant due to the raw amount of data required for all the textures in a game. If the video cards ram gets exceeded and has to swap textures often. This is referred to as 'thrashing' and is manifested by stutters, pauses in the game, slow load times (the appearance of white objects), just to name a few

Take a look at the differences in 512 and 1024. You will see very little difference. These screen shots were done at 1024x768 resolution. In order to really have any benefit at using 1024 textures, you need to be running 1280x1024 resolution, at the minimum.
This gets us into the resource problem most systems will have as the higher the resolution, the more resources are consumed, then add higher detailed textures, and the resources are really used up at a horrific pace.

Generally speaking, you would need a video card, with more than 768MB of video RAM, to be able to use 2048 textures effectively, and only with anti-aliasing disabled as anti-aliasing can quadruple the amount of video ram needed for any given frame to be drawn.