I used to use LXC maybe 5 years ago but I've since replaced everything with docker/compose. The main difference between LXC and Docker is that LXC is meant to be more like a Virtual Machine than a container. LXC containers run their own instance of systemd and can run multiple processes easily. Docker is meant to run a single process although people sometimes do hacks with supervisord or s6 overlay to run multiple processes.
At the time LXC didn't really have a concept of images like Docker, it was just base images like Ubuntu 18.04 or Debian 9 and you'd shell in the container and install your stuff.
LXD is a tool built on top of LXC, confusingly enough the LXD client is called lxc... It's higher level and might have the ability to use images, not sure, I never felt the need to learn it.
I tried replacing some components of my NAS server that were on docker/docker-compose with podman but unfortunately it was not a 100% drop-in replacement. I had networking issues in podman that I did not have in docker.
The network stack is implemented quite differently in podman than in docker, once you start using more advanced features the backward compatibility disappears.
Since it came second, I think it has a lot of technical advantages, avoiding docker's mistakes and what not. In the long term I'll probably switch to it, unless Redhat keeps shooting itself in the foot...