(EDIT: This is a traditional set of rules for "due to" and "because of", but there is disagreement over whether these rules apply to modern English. See further discussion below.)
They are not interchangeable.
He was lost because of the storm. (correct)
*He was lost due to the storm. (incorrect)
*He lost his way due to the storm. (incorrect)
He lost his way because of the storm. (correct)
These examples highlight the difference between "due to" and "because of":
He failed because of bad planning.
His failure was due to bad planning.
In short, "because of" modifies a verb, but "due to" modifies a noun (or pronoun). In common usage, though, you will often hear/see them being used interchangeably. More detail can be found in this article.
EDIT: See also this article, which mentions that
- "due to" is generally interchangeable with "caused by"
- "because of" is generally interchangeable with "on account of"
EDIT: Grammar Girl discusses "due to" in an article with references to Strunk & White, Fowler's Modern English Usage, and The American Heritage Guide to Contemporary Usage and Style, and my paraphrase of her conclusion is that traditional restrictions on "due to" are being increasingly abandoned by modern style guides and may eventually be abolished altogether.
Your example:
Those changes need to be made but the plan to make those changes does not yet exist.
is grammatically correct. The implication here is that the plan is expected to exist at some point in the future, but currently does not exist -- perhaps it is the next item on the work schedule.
Your second example:
The plan to make those changes has not yet existed.
sounds quite strained. It seems almost like a line from The Hitchhiker's Guide To The Galaxy or any other text dealing with time-travel, and would not find a home in natural English conversation.
If you wished to make use of still, you could indeed say:
The plan to make those changes still does not exist
or, phrased more naturally:
We still don't have a plan as to how to make those changes
In either of these two cases, as @PeterShor notes, the implication is that you expected the plan to be ready by this point, but for some reason it is not.
Best Answer
No. What first comes to my mind is that, even though they are very similar, yet connotes contradiction, whereas still carries the notion of defiance.
Let me view it from a different aspect.
Yet entails an element of surprise by the contrasting circumstances. Yet is like saying to your chum: "So, yeah, that's all nice and good... but check this out."
Still is more in the vein of: "So, yeah, that's that... but never you mind that."
It's a tiny difference, but it's a difference in the level of deductive capabilities that you, as the author, expect your audience to have. If they sense you consider them not capable enough of figuring out things without your solicitious guidance, and that you therefore opt for "yet" (... —Oh, yeah? What could it be?) instead of "still" (... —Yeah, yeah, go on, I follow you.), they might turn their backs on you. And stop reading your story.