Is either more stable than the other? How about caps vs. shotgun primers?

Not sure about stability, but Triple Seven produces less fouling and is easier to clean off.  My black powder friends say that the residue is hydrophilic and will rust your gun if not removed quickly.  Also, much easier to locate black powder substitutes vs real black powder, at least everywhere I've lived.

As for ignition, I've got musket caps, #11s, and 209 primers (I've even got Wincherster 209 Triple Seven primers) and they all still go bang after over ten years.

Never had a misfire with a 209 or musket cap.  I've had three FTF using #11s, once on game.  Weirdly, seems like #11s produce more muzzle velocity.  Maybe due to less blowback from the smaller orifice in the breech plug, IDK.

Pyrodex spoiled a deer hunt for me. Misty mornings caused powder to call to light on opening day and at the end of day unloading each day. Had to bum a pinch of BP to get the gun unloaded. Guys running BP me never had a problem.

Original Civil War era guns, cannon balls, etc are frequently found loaded with 160+ year old black powder that still goes bang.
A couple years ago friend of mine gave me a can Goex FFFg black powder that was dated 1985. It works just fine.

   I don't think Triple seven will last that long. I have some from early 2000's that won't go bang. It was kept in the house most of those years.

Other than being able to find pyrodex in stores, there is no advantage to it over real BP. Yes BP fowling is hydroscopic, but tap water and light oil solves that problem. Pyrodex has perchlorates which are nasty in their own right and harder to clean properly.

Substitutes are just that...substitutes for the real thing.  Stick to real black powder whenever possible.

Caps vs shotgun primers.

Depends what you want to do. Inline = shotgun primers.
Traditional = caps or flint.

I'm a traditional shooter, so caps for me. I have no interest in Inline rifles. Now if we had a dedicate BP deer season, I might invest in one. Our BP season falls into Alternative Methods Season where AR15 pistols are legal to use in Missouri.  

I've seen some Ruger Old Army conversion cylinders out of the UK that take shotgun primers. They are expensive and not really worth it to me here in the USA.

Based on past experience, the last portion of Pyrodex out of a can is mostly powder, which can not be consistent.  I think it is more corrosive as well.

The first ML I ever tried was a Thompson .50 cal sidelock loaded with Pyrodex. The powder had been opened the previous year. I recall pulling the trigger, no boom and as I turned my head away with a questioning look at the owner, the thing finally hang fired. Lag was about four seconds. Luckily I kept the muzzle down range.

When I later bought my own rifle, I used real black. Never experienced one misfire or hang fire.

In my experience black powder works better and cleans easier than Pyrodex.  I have been given old, half full, cans of black powder and it works fine.  I've read that somewhere there are samples of civil war era BP that are stored in vessels underwater.  No air and holds a more stable temperature.  When they are pulled and tested the powder still functions just fine.

I don't know if there's a difference in the priming compound between a cap or a 209 primer.  I've shot many old tins of caps that were given to me.