The two technologies appear to have similar functions. Initially I had a hard time wrapping my head around why to use Docker over LXC which to me is much more flexible and I don't lose ssh access.
However, interestingly enough as I used LXC more, I started to appreciate the spirit of Docker's approach to being so restrictive. From the view of writing applications and operating them:
It's a fundamental mind shift. If forces both Ops and Developers into a repeatable automated approach to building servers and coding applications. This shifts work instead to using good orchestration software (which I am in process of researching). The Docker base OS template is pared down to a single app environment and does not endorse an init, services, daemons, syslog, cron or running multiple applications.
Again though, I'll note all these feature is possible with LXC and by the way (with a slightly different spin) happens with Cloud Foundry powered by Docker.
You can run Dockers inside of Linux containers with virtually no performance impact.
Where I use LXC is for long running systems and creating isolation density within the Cloud. For example, a bunch of LXC's inside of an Azure VM which further separate my Docker apps separates my apps (maybe running inside of Docker) btw PreProd and Production.
Which one to learn first? If you want to really understand things, learn LXC and you'll know Docker. If using Cloud Foundry, learn LXC too. It's like starting with DOS then learning Windows.
Is one better than the other. No, they have different uses (which I'll outline later here).
See my hosting page for Virtualization technologies.