The logic in this article is egregious:

1. "Closely related individuals would have mated, allowing harmful mutations to accumulate in these groups."

At equilibrium, the expected fitness average of the population is constant. So harmful mutations don't accumulate, they stay at an expected constant ratio.

2. "The researchers conclude that Neanderthals were roughly 40 percent less fit than modern humans, meaning they were less likely to produce offspring."

The second part of the quote above purports to elaborate on the first part (", meaning"), but it fails to do so. What does "40 percent less fit" mean? Where did this number come from? The source doesn't seem to justify the number either.

3. "The regions of DNA around our genes"

What does that mean? The regions immediately next to genes? Or all non-coding DNA?

4. "Indeed, the influx of fresh DNA that came from interbreeding with modern humans likely slowed the Neanderthals' decline."

This sentence is wrong on several levels. First, since the first quote I cited is false, there is no mentioned support of a "Neanderthals' decline". The source does not seem to make any claim about Neanderthal fitness decline either.

What the article fails to mention is that the main reason why Neanderthal genetic material would be harmful to modern humans and vice versa is that the two lineages evolved independently for a significant amount of time, so that even a gene providing an advantage to a Neanderthal might be disadvantageous to a modern human.

The immediate expected consequence of interbreeding is a lowered fitness, because of partial incompatibility between the two different 'programs' being mixed. Additionally, modern humans were less adapted to local conditions, so even beyond incompatibility, some of their previously fine genes would be disadvantageous in Neanderthal's environment. Only in the long term could the gene influx have been advantageous to them, after a lengthy selective weeding process, and provided a large enough population existed to support it.

FYI: In smaller populations, consanguinity is more important. However, this doesn't directly harm a population at equilibrium, since the increased selective pressure against recessive harmful mutations will keep their number lower. On the other hand, more mutations make it into the gene pool simply through drift (selection is relatively noisier), so that expected average fitness is indeed lower.