The major difference is one is hard-wired and USB powered and the other is wireless and powered by a pair of AA batteries. Looking at both manuals they function the same and changing modes is the same.
Per the manual the gamepad should initialize in Xinput more (LEDs 1 & 2 on).
This is supposedly the mode we need for the Onefinity controller (RPi) connection.
To change modes press and hold the Home button for 5 seconds to jump to the next mode. Each subsequent press and hold of the home button will cycle to the next mode in this order:
The manual gives no explanation as to the differences between the modes.
For me, my ESM-9013 gamepad initializes in Android mode with LEDs 3&4 lit. I cycle through the modes to get to Xinput mode and everything works as it should as far as controlling X,Y, and Z on the CNC. When the gamepad goes to sleep (which the manual does not talk about at all) and I then want to use it so press any button to wake it up it reverts back to Android mode (LEDs 3&4) and I have to change modes again back to Xinput mode. But if you hold the button down a little too long it jumps right past the next mode and goes to 1&3 or 1&4, which means you have to try again to get to the default mode.
Here are the relevant pages from the ESM-9013 manual enlarged to a readable size.
The one I got, the EMS-9110 is a wireless model as well.
I found the manual on their website, it shows LEDs 1 and 4 should be on for Xinput mode. Which is default But nothing works in that mode on mine if I change it to 2 of the other modes, then x and y work, but the speed buttons are all mis-mapped, and Z does nothing.
I think I will just return it and order the correct one. Thank you for your help, it got me closer.