I can’t find it now but someone posted a really good chart that compared Jellyfin\Emby\Plex mostly around offline and customization aspects - it was minimal but had some nice references and maybe someone else remembers it and can find it for us. 
I kinda have a “long post” habit so I don’t mind laying this out but it’s really just my own perspective and you’ll definitely find lots of reviews and videos and such that can deep dive into more specifics between the three different platforms (I’ll skip Kodi or Infuse type options).
One aspect that is imperative is that I paid for a lifetime plex pass a LONG time ago. I have no financial consideration anymore. Plex isn’t getting any money from me and Plex isn’t considered a “cost” to use anymore; I got my money’s worth years ago. Jellyfin is free and fully functional while Emby isn’t so right there might be the entire decision made for some folks.
If you - or others - weren’t aware, Jellyfin is a fork of Emby. Emby went a bit proprietary and Jellyfin is an offshoot that stayed open source. Emby is pay - similar to Plex, you can use it for free but some functions are behind a paywall - and Jellyfin is free. What that means for those two projects is that foundationally they work very similarly. Emby has been around a long time, it’s just been long overshadowed by Plex’s dominance.
From my perspective both Emby and Jellyfin are very much solid replacements for Plex. What you’ll find as advantages will depend a lot on what you value in your setup. So, here’s my basic setup and why I still prefer Plex… for now.
I have 3 video libraries (Movies, TV Shows, Anime Shows) and two music libraries (Christmas Music, Main). I have all discovery, online services, friends, and rating stuff disabled at my account level. I share my Plex with a couple friends and my household - who aren’t always tech savvy - that also have that stuff turned off (I give them instructions). That simple setup - particularly only a few libraries - makes a lot of my Plex uses easier to manage through a lot of their recent changes. Plex doesn’t expect people to use libraries for organization or management anymore because of some of their feature updates for managing versions and editions and sharing so they’ve reversed that recommendation and expect as few libraries as possible so Plex features\functions have been designed with that in mind. Of course they didn’t really publicize that very well but that’s a different long talk.
At this point Emby, Jellyfin and Plex would work about the same for me and anybody else using it similarly just wanting to browse and play their videos.
Here’s where it gets different for me. For every 4k movie I keep an SD version as well (doesn’t require transcoding - a future proofing compatibility thing more than anything else). In Emby and Jellyfin I’d have to manage that with very specific file naming to manually differentiate those files and make sure they are listed under the same title but in Plex, it does it automatically even if the file is named pretty loosely. I also keep editions of some titles - directors cuts and such - and Plex now handles that as matched but separate titles with tags (a bit more manual but still mostly automatic) which I prefer to Emby\Jellyfin keeping versions and editions in the same title. If someone hits “play” on a title Plex will read the client information and if it’s a 1080p TV it’ll play the 1080p version, not try to transcode the 4kHDR version; it does this automatically. Emby\Jellyfin don’t do that very well and expect you to pick the right version to play. Emby\Jellyfin also handles editions the same way - manual selection from a dropdown mixed in with versions based on specific file naming. None of that is wrong but I’ve been able to be lazy with Plex doing “Just works” for most people when they need to play a file without having to think of which version or edition.
A note about file naming - it’s not draconian for Emby and Jellyfin. It’s just particular and significantly more so if you have different versions because of functionality built into reading the file name rather than just the media info. If you don’t have different versions or editions it’s pretty forgiving.
Another aspect is Smart Collections. Emby and Jellyfin don’t really do this yet (Emby v5 is supposed to get it). I like my smart collections because I’ve customized my Home Screen with them. I only have a couple default rows on the Home Screen and the rest I’ve replaced for me and my users (including on Recommend screens though I barely ever use that screen). For example, I replaced the Recently Added Movies and Recently Released Movies with a custom ones that removes watched titles so as someone watches movies that were recently added or released they get removed from the row (why keep one you’ve already watched in a “recent” row?). I also have a few collections that use random function to bubble up nostalgic action\adventure movies, movies that haven’t been watched in a long time, and some other generic ones. Usually about 20-30 titles.
The only default ones I left are for recently added TV Shows because that’s a special row that will group episodes and seasons and you can’t replicate that with a smart collection. I’ve tried because I wanted to remove watched items from there like I do movies, and even posted about it to try and work it out, but just can’t because Plex has secret sauce in that row.
I never use the music libraries on TV - I use PlexAmp on my phone with a Sonos system (not great but it works fine - not Plex’s fault and PlexAmp is pretty awesome). So when Music was removed it didn’t impact me but I still argued against it because most people’s best audio setup is their home theater setup now, not a receiver and box speakers anymore but a soundbar and subwoofer connected to their TV. Unless you’re an audiophile.
What that Home Screen customization - and the version\edition simplicity - does for me and my users is remove decision fatigue - browsing 1000 titles for “what do I wanna watch” - and allows them to just stay on the Home Screen and avoid extra navigation while being fed a selection of titles. Titles that if they browse for a bit will reset with a few new titles. This is a big reason the New Experience doesn’t impact me quite as much because I almost never actually browse away from the Home Screen.
I also have a lot of years of watch history and some of those collections are based on watch history so losing that by switching would be an annoyance.
Emby and Jellyfin cannot replicate that experience. They can’t do dynamic collections and you can’t set collections\playlists as rows on the Home Screen (you can customize a bit but not quite to the same extent). Or at least you couldn’t last time I fiddled with both platforms. I know Emby is getting that feature - or something like it - in their V5 release that’s coming up in the near future.
Another aspect that differentiates Emby and Jellyfin is application polish and ease of use. Plex pretty much had them beat for the most part. Emby has a lot more native client and server options compared to Jellyfin. Emby has an official QNAP and Synology package and keep clients up to date. Jellyfin doesn’t have as much native NAS support (there are projects that compile for QNAP and Synology) but does have good docker setup; but docker setup being easy once you get used to it still has a hurdle and isn’t quite as slick as native can be (hardware transcoding and network routing can be funny with docker - not hard but still an extra technical step to setup and manage). Jellyfin client support is also pretty mismatched - it works fine but Roku in particular was way behind. Jellyfin is very much an open source project and expects a lot of “you choose your settings”. Emby does too, when compared to Plex, but not quite as specific. Plex really is mostly a set and forget for most aspects of both server and client. And for both Jellyfin and Emby who share similar interface it just feels kinda older and just not quite so slick. Plex had them beat there - at least until now. Plex has added friction to local media access and lost ground to Emby and Jellyfin because of it. Like I said, I don’t mind because I barely leave the Home Screen but I understand why it’s particularly frustrating to others with more libraries (either personal or shared). Now, one benefit some of that extra technical setting exposure does do is that Emby and Jellyfin beat Plex in how customized and tweaked you can get things - particularly Jellyfin. If you like to tinker and fiddle and manually manage more things then they beat Plex hands down. On the other side, I’m not thrilled with managing another “project” … I did that with XBMC and other media solutions back in the day (custom ReplayTV firmware and a plethora of old desktops\laptops with dongles to attach to TV aux inputs) and don’t really wanna have to do that again.
Lastly, one area Emby\Jellyfin beat Plex is offline access. You can use both without an internet connection. This can be very important to many people (and is becoming a bit more important to me now that Plex isn’t just acting as a relay but is collecting data from it). One disadvantage is that remote access is a little tricker. Emby has a relay type option if you pay but if you’re used to old school media center setups or self hosting things then really it’s just a matter of proper port forwarding and DDNS setup (better yet, tail scale or VPN if you wanna harden it more but some clients won’t support that).
So… I guess generally I view Plex as still “better” than other options because it provides some specific QoL functions that I’d prefer not to do without if I can avoid it and switching services would be a chore (particularly file renaming - even with file renamer apps). As long as I can disable all those online services and discovery features as well. Because of that “chore” involved in switching, Plex has to get particularly annoying for me - and my couple of users - to switch. For others though … nothing stops you from running all three at the same time actually and trying them out; they won’t conflict.
The question was more which one is a successor though and mostly I’ve talked about Plex vs Emby\Jellyfin to provide a bit of context for my decisions. Personally I’ll be going with Emby when\if I switch just because it’s more native and polished and I don’t have a problem paying for a lifetime pass. I like the idea of open source projects and support many of them - and glad Jellyfin is there as competition - but I’m also a bit tired of fiddling and tweaking after having done it for a long time already. Emby is just further ahead because they have more resources available to them. Plus, if Emby do end up going the enshittification route switching from Emby to Jellyfin will be MUCH easier than switching from Plex.
Oh, and both Emby and Jellyfin forums are pretty good. They are very responsive. Feels like “old Plex” again there as well. I don’t really blame the plex reps\employees here for that change (I think I understand what’s happened on their end, mostly) but I can blame Plex as a whole for it - and do. A tip though - don’t mix and match between them. The two teams don’t really get along and they make it clear they are separate projects (Jellyfin is a bit more touchy about it than Emby).
Lastly, I just remembered that Plex does updates faster than Emby or Jellyfin do. In some ways that’s good and in some ways that’s bad so not really a plus or minus but something to consider.
Edit: I forgot to mention something. Plex is possibly going the route of enshittification. All companies try to make profit and some aspects of Plex’s direction kinda makes sense as a business wanting to stay afloat but many more aspects certainly feel like they’re leveraging their legacy of user\community good will for profits rather than living up to it or honoring it as a fair exchange. That aspect has me more inclined to switch than any of the technical or functional aspects.
Edit2: This is the Jellyfin QNAP build I’ve used: GitHub - pdulvp/jellyfin-qnap: Jellyfin server packaging for QNAP NAS ;; you can add their “store” to QNAP for automatic updates too.
Emby’s subscription model and handshake sounds like the data-mining camel has it’s nose in their tent as well. Privacy has moved to the top of my online concerns in the current political climate. Nufsaid.