I recently started working in a related field (RNA sequencing) and none of the other responses have really captured "what" it is that's going on here - and that's kinda necessary for the "why". When we say that the human genome project sequenced the human genome and completed in 2001/2002, what they mean is that there are databases with 23 sequences of A,C,T,G's for each of the 23 chromosomes plus another for the mitrochondrial DNA. This is "half" the DNA for a single person - or more accurately it was a composite of various parts of a couple people, in the case of the human genome project. It's only half because it's just one of the two sets of chromosomes that we get from our parents. It's not the genetic code of every person out there, just of one "reference" one. Variation within a species is quite low, so this is still extremely useful: chances are my genome looks a lot like the reference genome so if you want to talk about my genome, you can talk about it in terms of the reference genome. We can say that I have a SNP (single nucleotide polymorphism) where my DNA differs from yours in a single nucleotide (like switching an A for a C). It gives us a framework for talking about the genetic code of individuals of the species - we can say that the protein that does X is encoded in the gene at position Y on chromosome 12.

You can view the human genome at UCSC genome browser [1] and zoom in to the "base" level to see the actual sequence. It has lots of "annotations" where projects have marked where genes are (most of the genetic code does not encode proteins, genes are the parts that do, the rest is less well understood). You can see data there that shows you which parts of the genome vary from one species to another (generally mammals will be very similar to us in genes and less so between genes where the DNA is "less important" and so changes there don't result in death, so looking at where things are "conserved" by evolution tells you where the most fragile and important parts of the genome are, to a first guess).

What "they don't tell you" is that it's still not really complete. For example, there are bits of genetic sequence that we know go somewhere in the genome, but we don't know where. The way we sequence DNA, you only get to read little pieces at once and then have to string them together to reconstruct the whole thing. But one problem is repeats: sometimes the genome just goes "ATATATATATATATAT...." and if there are more there than the length of each piece you can read, it becomes impossible to know how many there are in total. Ambiguities like this exist because our DNA isn't just random strings of data, it has lots of structure that repeats from one place to another. So it can be impossible to tell where exactly a piece of DNA belongs, given the limited information we have. Also the annotations of where genes are is very much not finalized.

There are other problems with the sequences. For example, they'll include "N" to mean "some base, we don't know which" - there are millions of N's at the start of the chromosomes I've looked at. In reality, we all have some actual code there, but it's never been sequenced for anyone. Other structures confuse us: there is DNA encoding for ribosomal RNA which is used in ribosomes which are used to go from genetic code to actual proteins. So your body needs lots of ribosomes to keep everything being made so it needs lots of copies of the ribosomal DNA to produce enough ribosomal RNA - like 500 copies in a row. And it has those not just in one chromosome but in multiple. But the human genome project doesn't show it like that as it really should be: it has about half of one copy in the right spot. And then the rest of it is just spread in short bits throughout the rest of the places the code occurs. This causes problems if you are doing research and don't know what it is you're looking at, since it looks out of place. It's a complete mess. But despite that, it's good enough to do so much with.

Here's one use for a fully sequenced genome: RNA sequencing (which, I'll reiterate, I just started working on this a month ago so I've probably said at least one false thing already). RNA is made from DNA and proteins are made from RNA. So if you want to know what proteins are in a cell - and they're the things that actually do the work - then you can look at what RNA is present. This is easier these days than checking the proteins themselves. But RNA on its own doesn't tell you a lot, so you "align" the RNA to a reference genome: given a little piece of RNA (like 100 nucleotide bases long), where does it fit in the genome? It often has a unique spot it could have come from, so now you know what DNA was turned into what RNA. If there's a gene there, then you know what gene is being "expressed," meaning turned into protein via RNA. Genes don't do much if they're not expressed, so this is very useful to know to understand what is happening in a cell. Looking at RNA sequencing versus DNA sequencing is like looking at what software is running on your computer versus looking at what code is stored on the hard disk. Both tell you a lot and understanding what code is on the hard disk will help you understand what you're looking at when you're looking at what code is running, like "Oh, these operations belongs to the bash executable - so bash must be running".

[1] https://genome.ucsc.edu/cgi-bin/hgGateway (hit GO if you just want a piece of human DNA, hit "base" button in zoom in if you want to see actual sequences but that's too close to see much of the structure that the annotations provide)