OK I'll take a punt here. I say I punt because I have never held the items in question in my hand but I'm a tech by trade and have been flying R/C and using Lithium cells for 27 years, I have half a clue anyway.
I would guess the difference is this, 3rd Party batteries will be totally that, 3rd party. Every component in them will be from an unknown manufacturer and you can remind yourself that you are flying on batteries constructed on parts supplied by the lowest bidder ... I wouldn't use one personally.
What makes DJI batteries different from the run of the mill LiPo is the onboard controller board. Depending on model it will control charging, cell balancing, short circuit and overload protection, storage discharge and other functions. I can can tell you (as a Technician AND as an R/C instructor) I was dubious about DJI's claims about their controller boards when I first started flying U.A.V.s but have been won over as time has gone by. These controller boards are the difference between an OEM battery and a 3rd party.
LiPos while certainly volatile are not black magic but they need to be handled and operated carefully for maximum life, safety and performance. My charger for my "dumb" R/C LiPos cost $600 and is worth every cent because it is totally configurable and balances and charges intelligently while monitoring the battery status. DJI have kindly included this circuitry onboard in each unit (hence the high cost) and allow you to use a dumb charger.
Dead battery packs are easily repacked .. just swap out the old cells for new ones, my best guess is that "refurbished" DJI packs will have had the old cells replaced and yes, probably with cells not to the standard of the OEM cells but will still be using the original DJI onboard charger controllers.
As I said, I have no proof .. perhaps someone with a "refurbished pack" might like to do a little surgery and confirm but if what I suspect is correct the difference will be :-
1. OEM DJI battery packs = Selected cells from a reputable source with DJI's own onboard management software/hardware (my choice every time but I run a business and my insurance requires that)
2. "Refurbished DJI battery pack" = repacked with cells perhaps (probably) not to the same standard as the OEM packs but still using DJI's own onboard battery management hard/software. If you're hard up for cash and this is the difference between flying and not flying it's your next best option if (again if) it's the DJI onboard battery management. You may not get the same life out of the cells but charging and short/ overdraw protection and all the other safeguards should be still there for you.
3. "3rd party battery packs" = Cells sourced from who knows where controlled by unknown onboard management (if any). You may just be getting reject cells from a known manufacturer with a flashing LED circuit. I'd let my aircraft rot on the ground before using one.
Once again and for the 3rd and final time "THIS IS INFORMED SPECULATION USING MY OWN KNOWLEDGE OF THE INDUSTRY AND LOGICAL DEDUCTION NOTHING MORE."
Still if you're in the situation where you *have* to make this choice then it may arm you with the right question to ask the vendor.
For that matter, those of us who continue to fly increasingly old aircraft are eventually going to face a situation where there are NO genuine DJI battery packs. This thread has just made me decide to keep my old tired but genuine cells in cold storage (literally in the freezer) and get them repacked by a reputable local battery service when I can no longer obatin genuine parts. I should have a good store of them by then.
Regards
Dec