The main difference between the two transmitters it that they're used for different radio triggering systems. The YN-560-TX is for Yongnuo's manual-only 560/60x system; the YN-E3-RT is for Yongnuo's cloning of the Canon -RT radio triggering system built into Canon's latest speedlites.
My understanding is that ... the YN-E3-RT won't work by itself, as opposed to the YN560-TX.
This is incorrect. The YN-E3-RT can be used in-hand (off the camera hotshoe) to make setting adjustments and test fire off-camera flashes, just like a YN-560-TX.
... what is the functional/workflow difference between the two units?
They are essentially identical in workflow. You place them on the camera hotshoe, set it to the same channel as the receiver units (built into the flashes, or added onto the flash foot/cabled to strobes), and then you control the remote flashes via the interface of the transmitter and fire the flashes either with the transmitter's TEST button, or the camera's shutter button.
The difference is in the functionality of the triggering systems.
The YN-560-TX can only give you sync (firing the flashes) with RF-602/603/603II receivers/transceivers, sync and group on/off with RF-605 receivers, and sync, zoom, and power control over Yongnuo YN-560III/-560IV/-660/-740 speedlights.
The YN-E3-RT transmitter only works with the Canon -RT system and clones (which will be more expensive than manual-only Yongnuo triggers), but gives you nearly full control over the remote speedlights, as if they were on the camera hotshoe. You will have HSS, TTL, 2nd curtain, Gr mode control, etc. As well as a slough of other UI enhancements and features such as being able to use an -RT flash as a remote shutter release via the YN-E3-RT/camera hotshoe connection (if the camera is a post-2012 body) and four-digit ID codes to avoid radio interference from other shooters using the same system in your vicinity. There's a lot more control over the remote speedlights in this system.
Manual-only gear has the advantages of being universal to all camera brands and cheaper/simpler to produce, while TTL/HSS gear is (usually) dedicated to a single camera system and is much more difficult to reverse-engineer, because the communication between the camera and flash is much more complex.
There is also the fact that you can't really mix these two systems. While you can use a YN-600EX-RT II speedlight from a YN-560-TX, but functionally, it can't do anything more with that transmitter than a YN-560 IV could. And you can't trip a YN-560 IV from a YN-E3-RT, unless you put a YN-E3-RX on its foot, and it will still be a manual-only flash whose power cannot be controlled from the transmitter.
This is why the -RT gear will be more expensive. All the flashes have to have -RT built in if you want the full set of features the YN-E3-RT can offer.
Other brands do offer systems where TTL and manual-only gear can be mixed, but they won't offer the same breadth of features the -RT system does. Whether or not you need those features is up to you.