I use emby, but as I've paid for lifetime membership several years ago the bar for me to switch is rather high.
I'm not always up on the goings on with plex, but I understand many people were upset with Plex's decision to force you to have internet access to authenticate user accounts even for a local server. For most people, I doubt that is an issue in practice, but understandably the idea doesn't sit right with a lot of people. I'd read into that part first.
Media-wise they are both the same. Where you will see differences is supported platforms, connectivity options, app support, etc.
I have my main Emby server at my house, but I would love to be able to have satellite nodes to share the load. To my knowledge, neither emby nor plex can do that, but if Plex did it first I would consider switching.