I like Apt a lot, but it does have some shortcomings.
First, of course, it requires an apt-based distro, so software distributors need to have apt alongside all the other packaging alternatives. Or they just provide a bash script. Ew.
Second, apt doesn't elegantly handle different versions of the same package. That's rarely an issue for well-established C libraries, but it's a big issue for Java and most of the dynamic languages. So you end up with a host of language-specific package managers.
Third, there's stuff beyond simple files that falls outside the wheelhouse of apt. Networking, configuration, whatever. The paradigm for apt is very much to have a small number of systems, manually curated by dedicated sysadmins. When you start scaling up to tens, hundreds, and thousands of hosts, you end up writing and maintaining long scripts to initialize a freshly-installed system and put it in the right state. And those script will break as packages evolve. Getting a system into a known-working state is difficult.
A specific example: I installed and set up GitLab on a Debian system a while back, and it was a huge pain. It's not just a package, after all, it's web code, a couple daemons, a sql db, a redis db, git repos on the filesystem, and more. The install guide was pages and pages long. I never quite got it working right (something about SSL certs, IIRC one of the daemons wasn't using the system CAs?).
So I tried docker for the first time, and had GitLab up and running in about 10 minutes.
And if I ever wanted to migrate to a different host, spin up another node for load balancing, or do backups and restores, you bet your ass I'd use docker.
Apt is great for carefully curated, individual systems. It was perfect for the world circa, say, 2005, and the world would be better of if we'd all standardized on it then. But even if we had, somebody would've invented something like docker in the meantime, for managing complex software (like GitLab) on tens, hundreds, or thousands of hosts.
But docker has all the issues pointed out in the article above, and more besides (every image is hundreds of megs, because it contains a full, running Linux system...that's just crazy).
Nix can do the package management thing that apt does so well, and it can also do the reproduceable, holistic system build thing that Docker does. It can also make management of language dependencies (i.e. Java JARs) much more clean and elegant.
It's seriously worth checking out.