Avgas has a higher octane content than most pump gas – the octane rating isn't just because of tetraethyl lead. As for stability, it's probably better because of what's not in it. It's hard to find specific details, but oxygenates are the biggest culprits. Ethanol-based additives are the most problematic, but I'm pretty sure the additives in even ethanol-free gas reduce its stability. Avgas has a legal shelf life of one year, but I'd bet you won't find much pump gas that's sat unsold nearly that long, so it's not likely Avgas is fresher. I'm fairly certain premium gas will last longer on average, but more because the lower-quality batches of distillates are more likely to be used in "regular" gas. As I understand it, a higher octane rating doesn't contribute much to stability in and of itself.
Anyway, in my case, I always fill my tank with ethanol-free and stabilizer after the last ride, then run a bottle of cleaner in the first fill. I've never had any issues running the "old" gas. Now, the best practice is to fill the tank before storage, then drain and refill with fresh gas, but I don't see it as necessary. The worst thing to do is to drain the tank before storage, or even leave it half-full. That invites condensation, and any gas you didn't get out of the tank will oxidize rapidly. Ultimately, if you can get ethanol-free and it's stored properly, you can generally count on at least a year of storage life. Better safe than sorry if there are any doubts, but I've got a half-full tank in an old sled that's sat for five years, and it still smells fresh.