How should the popular press compare similarity of genomes?
1–4% is from an evolution point of view. 99.7% is from a sequence point of view. Because they are measuring different things, they can be compatible with each other. The correct interpretation is: 1–4% of a non-African genome is inherited from Neanderthals and the sequence of this 1–4% differs from modern human sequence by 0.3%. PS: so, for each non-African, Neanderthals contributed 1–4% * 0.3% of base pairs not present in modern human. This sounds tiny, but even a single SNP may lead to severe disease. This tiny fraction has known functional impact to diabetes and high-altitude adaption.



Others pointed out two related questions:


How many genes do we share with our mother?
Do apes and humans share 99% of DNA or 99% of genes? What is the difference?


These questions are really about the ambiguity in the definition of "gene". Importantly, traditionally in popular science and population genetics, a gene could mean an allele. This makes differences.

For example, by common uses of "gene" in biology, the answer to "how many genes do we share with our mother" would be nearly 100% – every human has nearly the same set of genes. However, clearly this is not what the asker wants. By gene, he/she really means allele. The expected answer is 50% – we inherit 50% of alleles from our mother.

In the end, what is a "gene" exactly? I don't know. Biologists can often figure out the meaning of word "gene" in the right context, but non-biologists will sometimes have troubles with the ambiguity.