Below is an answer I got from Mark (the creator of Apophysis) about the options in Render to Disk:

"This following used to be in the help file (for a long time it was the
  only topic there). It'll be back eventually...

  The Sample density parameter determines the number of times that the flame
  algorithm is iterated. This parameter has the most obvious effect on the
  quality of an image and images rendered using a low sample density will
  look very grainy. The exact value to use may depend on the flame that you
  are rendering: thin, wispy flames will need a higher sample density than
  tight, solid looking flames. The default value of 200 is a good place to
  start.

  Sample density has a large effect on rendering time: the higher the sample
  density the longer the image will take to calculate. It is adjusted
  automatically for image size and zoom factor, so larger images will take
  longer to render, as will images that have been zoomed in.

  The Filter radius parameter controls the strength of a blurring filter
  that is applied to the image. The filter works in conjunction with
  oversampling to anti-alias the image. The default value of 0.4 works well
  in many cases, but you can reduce it if you think the image needs to be
  sharper. With larger values the image will look more obviously blurred.
  The effect of a certain filter radius value might change with a change in
  the oversampling factor.

  The Oversample parameter controls the degree of oversampling used for a
  render. Oversampling means that the flame is calculated at a larger size
  than intended for the final image, and then resized down. Another way of
  looking at oversampling is to think of the image as being calculated on a
  finer grid than the pixel resolution. An oversample value of 1 means that
  no oversampling occurs, a value of 2 means that the flame is calculated at
  twice the size of the final image, and so on. Oversampling helps to add
  detail and anti-alias the image. The default value of 2 works well in most
  cases but you might try a higher value if aliasing is very obvious in the
  flame that you are rendering (this can be the case if it has very fine
  lines or straight edges). Increasing the Oversample value has a large
  effect on the amount of memory required to render an image."

  Mark

More advice from Mark:
My question was why it was going to take so long rendering a flame at 2560 x 2048 with the sample density at 2000.

"Yes the size combined with the sample density and the
number of trips if you're liming. You don't usually
have to go as high as 2000 for the sample density: if
it's a tight flame you won't see much difference from
a much lower density. Because the sample density is
scaled with the size of the image, you can do test
renders at a smaller size and see if the sample
density is high enough, and not have to change it when
you do a larger render."

 

Trying out the different settings for rendering.

PRESET: 640 x 480 PNG High Quality
(Images are resized to 500 x 375)

Default
Sample Density: 200
Filter Radius: 0.4
Oversample: 2


Sample Density: 200
Filter Radius: 0.4
Oversample: 4
 


Sample Density: 400
Filter Radius: 0.4
Oversample: 2


Sample Density: 400
Filter Radius: 0.4
Oversample: 4


Sample Density: 200
Filter Radius: 0.1
Oversample: 2


Sample Density: 200
Filter Radius: 0.2
Oversample: 2


Sample Density: 200
Filter Radius: 0.3
Oversample: 2