Do apes and humans share 99% of DNA or 99% of genes? What is the difference?
So, a quick molecular biology lesson. 


Proteins are the things that make up a good percentage of our cells (which make up a good percentage of us), and are the things that do the work of the cells - many are catalysts and are known as "enzymes". 
Proteins are encoded by genes - while the statement that one gene
codes for one protein is not quite correct (one gene can code for
different variations of the same basic protein), it's a good way to
think about things in this context.
Genes are made up of DNA, a polymeric molecule that constitutes our chromosomes, the informational portion of which resides four “letters” (chemical bases).
However, now we get to the key part — although all genes are made of DNA,
not all the DNA of chromosomes makes up genes. In fact, as @terdon    mentioned in a comment, only about 5% (or less) of the 4 billion letters in the total DNA — the genome — constitute genes - those sequences that
directly code for protein. 
The function of the rest of the genome is not entirely clear. Some is regulatory, some may be structural, and may be “junk DNA”. However it’s stuck      around for millions of years, so it we assume it must have some purpose. This non-coding DNA differs between species to a greater extent than the genes themselves do, so perhaps it somehow contributes to the differences between organisms.




From AndroidPenguin

Here are the links to a paper about the function of "junk" DNA from 2013. 


Summary in NY Times
Abstract in Nature
ENCODE threads on nature.com