Please keep in mind that you will invalidate any warranties by changing the firmware of your drive to anything unsupported.
I don’t know how you can find to make sure it is the same - try looking for user experiences. No real surefire way - but if you find images of the internals - check to see the chipset is the same at least.
If you do want to flash it - i think the flasher SFDWin is not very smart and if you start it up with a flasher correct for your firmware [if you do find one from Asus] and then you swap the file with the firmware in it for the firmware you want to flash [renaming that to the same name as the correct firmware] - that might work as I had once crossflashed a Ricoh to Samsung firmware.
Otherwise, you’ll need to see whether it uses a MediaTEK chipset, and whether MTKFlash works with it. There are tutorials around on how to use MTKFlash - to use it to backup your existing firmware so you can hopefully restore it later if something goes wrong - and also to force-flash a different firmware onto the drive.
I do NOT encourage you to proceed if you have any doubts. Are you sure you really need lightscribe? is it that important to you? can you live with a possibly dead drive with no recovery? I think these are the important salient points to think about before proceeding. As for whether they are the same - i have no clue - but the model numbers look radically different and that usually is the first thing i’d look for before trying.
Chef also brings up a valid point [you beat me to this] that drives to do LS have some additional hardware - and while the drives are “nearly” the same - say BenQ 1650 vs 1655 and LS firmware will run on a 1650 - the 1650 is unable to burn LS discs because it is missing the LS electronics.