I don't blame you for being confused -- there is no single standard for comparing how similar two genomes are, since it depends a lot on what kind of comparison you're making. Nucleotide-for-nucleotide, you are right that there are almost no differences between the protein-coding sections of Homo sapiens and Neanderthal genomes. I don't know the exact number off hand, but it's surely much closer to 100% identity than 99%.

However, when people talk about what % Neanderthal DNA a person has, it's usually talking about larger "chunks" of the DNA. This is a measure of heredity that is more akin to saying a person has 50% European ancestry and 50% Asian (or whatever). The European and Asian human genomes are almost 100% identical, but you could analyze that person's DNA to figure out what fraction of the DNA was likely to have come from European populations and what fraction was likely to come from Asian populations. A common way to do this is to look for single-nucleotide variants (called SNPs or "snips") that are common in one population but rare in others. If you find several such SNPs in a region of the genome, the probability is high that the person has heredity from that population.

Edit: Also, it gets even more complicated when you start taking into account non-protein-coding parts of the genome, which can be more variable in sequence and size, as well as situations where whole parts of the genome might have been duplicated.