Just to clarify definitions, your genome is made up of sequences of DNA. DNA is constructed pairs of four nucleic acids, or nucleotides (A,G,T,C). That DNA has many loci within it, each codes for a gene. Loci are given gene names such as SHH (sonic hedgehog) which is part of a discipline called gene nomenclature. Humans have two versions of each gene, one from each parent. These genes, such as SHH, have different variants called alleles. This is where you may have heard the terms homozygote and heterozygote, when a person has two copies of the same allele or two different ones respectively. You may remember talking about sickle cell anemia in high school.
So is it wrong then to say that apes and humans share 99% of their DNA or is it equally correct to say "genes"?
Well the similarity between humans and chimps in terms of known DNA sequence is about 98.8%, there is fairly low sequence divergence, so in my opinion you are right to say that they share 99% of their DNA. This means that in 100 nucleotides there is roughly one difference e.g. (showing 90 bases, made using random sample of four letters in R):
Human DNA strand:
atgactgtagcccatga c gtaaacgtaccaagcctcctcggctgtcccgaaatagatacgcctggtagacgtattaatagtgagtaa...cgt
Having said that, given that there is >3,000,000,000 base pairs in the human genome (the sequence of 4 nucleotides that make up the DNA) that equates to ~36,000,000 different base pairs (bnetween humans and chimps) which likely equates to a large difference. Sometimes even a single nucleotide polymorphism (SNP, pronounced snip), a change in just one base pair, can bring about quantifiable changes:
a single base mutation in the APOE (apolipoprotein E) gene is
associated with a higher risk for Alzheimer disease.
Here is a little reading on the humans vs chimp. You may also wish to see this similar post on biology SE.
But the editor of your original post is right... only the Euchromatic region has been sequenced.
"Humans and apes share 99% similarity in the coding sequences of their DNA"
The euchromatic region is often thought of as the protein coding part of the DNA, and makes up 92% of the genome (I think this is perhaps 92% of genes, but only a small part of the physical DNA).
The rest, the heterochromatin, contains non-sequenced information and could harbour some of the DNA variation between the two species. Some work has suggested that the heterochromatin, previously labelled as Junk DNA, actually has some effect on traits. For example in Drosophila there has been studies which show effects of the Y chromosome on traits despite the Y chromosome being mainly heterochromatic.
Further: I think a common source of confusion comes from the use of the word gene. It is often used interchangeably to mean either Allele or Genetic Locus. When someone says that humans and apes share 99% of their genes they mean they share 99% of their genetic loci. When someone says that a person shares 50% of their genes with a parent they mean that they share half of their alleles with that parent (it is only roughly half due to recombination and mutation). An individual has all (barring genetic defects such as deletions) the same loci as their parent.