Captains Log is (or started as) a standalone logging tool that did not require an account with a web site or any internet access at all. That was and is what sets it apart. If you just want to log your data without sending it to the internet for others to see, that's your tool.
ED Discovery or EDMC both sync data to EDSM, EDDN and EGO
ED Discovery gives you local access to your data but EDMC only sends it to the web without any interfaces for you to view, search or map locally.
EDSM is a terrific site and I highly recommend setting up an account and using ED Discovery.
You can access and share with others your flight logs,
ship details, inventory
and map of your travels.
You should only run one of these tools at a time. You don't want two or more syncing the same data.
Journal logging happens no matter what, even if you aren't running any of them. All of them will read your journal files after the fact, so they don't all need to be running while the game is running.
If you want scan data and jump information to be synced to EDSM and EDDN as you go, you'll want ED Discovery running.
If you want to use Captains Log, you can open that any time and the data will be there.
I used to use EDMC when I was in the bubble so it could send market data and ship outfitting data, and I would use ED Discovery when I was out of the bubble on a trip.
Now ED Discovery gathers and sends market data so I don't see any advantage for EDMC now.