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"It is common that people outside of science mistake the two numbers, and confuse and/or conflate the two. You are not the first to make this common mistake."

I see only one set of numbers reported here.
This report's language clearly equates the units of measure for SNPs, nucleotides and "base pairs":

"Nucleotide diversity is based on single mutations called single nucleotide polymorphisms (SNPs).
"The nucleotide diversity between humans is about 0.1%, which is 1 difference per 1,000 base pairs.[4][5][6]
"A difference of 1 in 1,000 nucleotides between two humans chosen at random amounts to approximately 3 million nucleotide differences since the human genome has about 3 billion nucleotides."

This report clearly says that human genetic diversity amounts to 3 million out of 3 billion base pairs, or .1%.

The report on Neanderthals says they are "more than 99.5 percent identical, differing by only about 3 million bases".

There is no suggestion here that they are talking "apples and oranges".
No explanation of how 3 million bases amongst humans is somehow different from 3 million bases between Neanderthals and humans.

In both cases the denominator must be our 3 billion base pairs, making the identical nucleotides 99.9%.

So, as far as I can see, this is only a discussion of "apples".
Now if you wish to introduce "oranges" into the discussion, then you will have to explain just what an "orange" is, how it differs from an "apple" and, most important, how that somehow changes the denominator of 3 billion base pairs to some other number -- a number which mathematically reduces the identical DNA of Neanderthals & humans to 99.5%.

This will be interesting... ;-)

"It is estimated that approximately 0.4% of the genomes of unrelated people typically differ with respect to copy number. When copy number variation is included, human to human genetic variation is estimated to be at least 0.5% (99.5% similarity)."

Ooooops, there's an "orange", showing human DNA only 99.5% identical. Wouldn't want to get that mixed up with those other "apples," would we?

The report that cites both the 99.5% difference and a 3 million base pair difference is possibly citing only the confirmed base pair differences over the amount of the neanderthal genome sequenced at that point.

Obviously a 0.5% difference over 3 billion base pairs is 15 million, not 3 million.

So how did that same source get both figures? They are obviously confused, or leaving out a critical bit of info.

Do you have a source for human variability similarly being 99.5%? The figures both you and I have cited have given it as 99.9%.

Either way, the data I have cited has given the human similarity as 99.9% and the neanderthal-human similarity as 99.5%. This is exactly what you would expect if the most distantly related human populations shared a common ancestor within the last 100,000 years, but neanderthals branched off some 500,000 years ago. Five times as much genetic difference, almost all of it in ‘neutral’ DNA regions of the genome.

Throwing in copy number variation when comparing humans but not doing the same when comparing neanderthals with humans would be an example of an ‘apples’ and ‘oranges’ comparison.

I notice that you got the latter part also from the same Wiki entry that gave the human similarity as 99.9%.

Human neanderthal similarity is 99.5%.

Five times as much difference, as one would expect from them branching off some 500,000 years ago, while all human populations share a common ancestor some 100,000 years ago.