Both are ways to manage containers, and both can use the same container runtime provider, IIRC.
They are different in how they manage the containers, with docker/docker-compose being suited for development or one-off services, and kubernetes being more suitable for running and managing a bunch of containers in production, across machines, etc.
Think of kubernetes as the pokemon evolution of docker.
I did come across it before, but it feels like just another layer of abstraction over k8s, and with a smaller ecosystem.
Also, I prefer terminal to web UI.
Setting up k8s with k3s is barely two commands.
Works out of the box without any further config.
Heck, even a multi-node cluster is pretty straightforward to setup. That's what we're using at work.
Several services are interlinked, and I want to share configs across services.
Docker doesn't provide a clean interface for separating and bundling network interfaces, storage, and containers like k8s.
I used docker for my homeserver for several years, but managing everything with a single docker compose file that I edit over SSH became too tiring, so I moved to kubernetes using k3s. Painless setup, and far easier to control and monitor remotely.
The learning curve is there, but I already use kubernetes at work.
It's way easier to setup routing and storage with k3s than juggling volumes was with docker, for starters.
Junji Ito having a braingasm rn