Here goes:
While Putin survived in power, I think what happened highlights the weaknesses of his regime more than any strengths.
To me, the most striking aspect of thiese events is how little armed resistance there was to Prigozhin's March on Moscow. The Wagner forces took over 2 major cities (Rostov and Voronezh) almost without a fight. It seems like there aren't many military units truly determined to fight for Putin.
This was NOT because the whole thing was a staged show, bloodless on both sides, as some conspiracy theorists claim. The Wagner forces showed they were more than willing to use force when challenged. They in fact killed about 15 government troops (mostly crew of planes and helicopters they shot down).
While there was no great desire to fight for Putin, there also wasn't much outpouring of support for Prigozhin. Only a few government troops actually switched sides, as opposed to just standing by and watching. This may be why Prigozhin ultimately decided to make a deal, rather than press on. He couldn't prevail with just the troops he already had, and wasn't attracting enough defectors.
I wouldn't make too much of the now-famous video of Rostov residents cheering the Wagner forces, as they left. Rostov is a city of 1 million people, and it seemed like there were no more than a few hundred people cheering. Plus, Rostov residents who aren't big fans of Wagner/Prigozhin were probably keeping quiet (for obvious reasons). That said, it's also notable there was no great outpouring of popular support for Putin in Moscow or other cities. Few were eager to go into the streets to show their support for the great Czar of the Russian World.
The bottom line here is that military, elite, and popular support for Putin is pretty thin. That creates opportunities for future coups and/or uprisings. A general or politician with more appeal and/or a better plan than Prigozhin could succeed where the latter (partially) failed.
Former US Ambassador to Russia Michael McFaul makes a good point: There is this theory that Putin never concedes and always responds to pressure by escalation. Yet yesterday, he went from denouncing the coup plotters as traitors and promising dire punishment to amnestying them all, and apparently conceding that Wagner will continue to have an independent existence (though possibly not under Prigozhin, who apparently agreed to exile to Belarus). That's quite a comedown for Putin, and an obvious sign of weakness. When facing the possibility of defeat, Putin cut a deal rather than stuck to his guns. As McFaul argues, it strengthens the case for supporting Ukraine to the max.
The deal Prigozhin and Putin struck seems good for Wagner as a whole, but bad for Prigozhin personally. If the latter is going into exile in Belarus, that means he will be totally at Putin's mercy. Belarus dictator Alexander Lukashenko is little more than Putin's puppet, at this point. So long as Prigozhin is in Belarus, Putin can off him any time he wants. And he's not known to be a very forgiving sort of guy, inclined to let bygones be bygones. As Darth Vader might put it, "Putin is not as forgiving as I am." However, perhaps there are aspects of the deal that aren't publicly known and/or Prigozhin doesn't really mean to go to Belarus. He hasn't been seen or heard from since yesterday.
In his speech yesterday, Putin analogized the situation to 1917. If taken seriously, this analogy puts him in the role of Nicholas II, or perhaps Kerensky. I would add that it's worth remembering that Kerensky survived 2 coup attempts before being overthrown by the Bolsheviks in November. He overcame the first Bolshevik attempt at a takeover in July (at which time he made a terrible mistake in not executing the Bolshevik leaders), and then General Kornilov's coup. There are many differences between Kerensky and Putin (including that the latter has been in power much longer and leads a far more brutal and illiberal regime). But I nonetheless suspect this won't be the last serious challenge to Putin's rule.
In sum, Prigozhin's coup attempt and the deal that ended it highlight weaknesses of Putin's regime. That does NOT mean he will definitely be overthrown. He still has some sources of strength. But it does mean he's much more vulnerable than many thought before the events of the last 2 days. Prigozhin could turn out to be Putin's Kornilov. If so, I hope whoever plays the role of Lenin installs a less horrific regime than the original.