eeprom is the entire file containing all 60 models, and the general settings.
Flash memory is the firmware resident in the transmitter.
You set up the eeprom with your models and settings.
The programmers create the firmware that you flash.
For Reading or Writing Eeprom you must have the Taranis ON before connecting to the USB on the computer.
Write EEPROM to TX / Read EEPROM from TX
Write will load the currently open and active model setup (eeprom) to the transmitter. Read will download the current eeprom from the transmitter to an open file.
Write EEPROM memory from file / Read EEPROM memory to file
This is the same as above, except you write from a saved eeprom on a disk or write the eeprom in the Tx directly to a file on the disk.
For Reading or writing Flash memory you must have the transmitter off before connected with USB cable and remaining off while doing the flashing.
Write Flash memory / Read flash memory
Flash memory is the current firmware - I've never had occasion to read flash memory (to a file) I've always use write flash memory to write or "burn" the current firmware. It's safe to say that the first wave of Taranis users will uncover a few more minor bugs in the firmware that will need to be remedied. The good news is that the developers are very responsive to bug squashing and the other good news is that its very unlikely there will be any bugs of consequence.
Companion9x is used for the 9x transmitters in a variety of different flavors as well as for the Taranis. This is the reason they held on to the 9x moniker.
With the Taranis:
The Taranis will save the eeprom in the eeprom backup folder (user defined location in preferences) as an .eepe file. This file can be opened for editing or whatever.
You can save your eeprom as either .eepe or .bin. I've been doing both just for testing -- .bin is a smaller file, but I don't see any difference other than that. I suspect that the ability to save as .eepe is a hold over from the 9x and the additional size of the file is related.
In practice - I have been saving my model setups as .bin That's my recommendation unless some other information comes to light.
I've not found .hex to be relevant to the Taranis.
New firmware will download as a .zip. The .bin file inside such the current opentx-taranis-en-r2523.bin are the firmware files you burn to memory.
Finally, there are limited changes you can make to the firmware files. Specifically you can change the splash screen using the Customize your Tx in the Burn dropdown menu. Here you load the firmware, then load a new image and save the firmware with the new image replacing the existing splash screen.
Attached is a splash screen that yesterday I put on the Taranis I'm testing.
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