The main difference is LXC boots an init in the namespace as the first process so you get a standard OS environment with multi-process support. Docker runs the application directly so you get a single process environment, unless you decide to run a process manager. IMHO this creates far more problems than it solves as anything nonstandard would, as you now need workarounds for basics like logging, networking, daemons and anything that expects a standard OS environment
Docker also uses layers to build containers and ephemeral data. None of these 3 critical decisions have seen much technical scrutiny resulting in persistent confusion in the ecosystem and with scrutiny many users will find they can add far more management overhead and complexity to containers. We have written about some of these issues here. [1]
For instance OP is referring to unprivileged containers and user namespaces that let you run containers as a user process. LXC has supported these since 2013. Docker added user namespaces support in recent versions. However taking just one issue - because of the use of layers to build containers you can't use unprivileged containers for app builds as only root users can mount filesystems. So a lot of these decisions have all kinds of tradeoffs that should be better known and discussed. [2]
Disclaimer: Founder at Flockport and trying to build a simpler alternative with LXC with support for app builds, provisioning, orchestration, service discovery and an app store. [3]
[1] https://www.flockport.com/guides/say-yes-to-containers.html
[2] https://www.flockport.com/guides/understanding-layers.html