For one, consider how much really needs to be the same. All of the work that cells do needs to be there. How to make the endoplasmic reticulum and other organelles is the same. Heck, our genome and a banana’s are about 50% the same. And the Neanderthals need to have the same body plan, bones, heart, liver, teeth, etc. The genes for building these things are going to be virtually identical–I expect them to be 100% the same.
But it’s how these genes are turned on and off that’s interesting. Most of our DNA isn’t in genes. We used to call it junk DNA. Now we’ve learned that there are things in the DNA that turn genes on and off. And when you turn a gene on can greatly affect the structure of the organs involved. Darwin’s Galapagos finches had different beaks depending on what kind of food they ate. But the genes for the beaks were the same. So how did they have different beaks? The genes were turned on at different stages of development while in the egg. If you turn the gene on sooner, you get a larger, stronger beak, good for cracking nuts open. Turn it on later, and you get a narrow, pointy beak, good for picking insects out of plants.
So, the genes for us and Neanderthals are mostly identical. It’s the switches that turn these genes on and off that’s different.