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https://civilization.fandom.com/wiki/Conquered_cities_(Civ5)
Short answer.
They're basically two different versions of the same thing.
Puppet means you can't pick what the city does or builds even once any resistance has been stamped out, but there is less of an unhappiness penalty.
Annex means that after all resistance has been stamped out you can directly tell it what to do, but at a cost of more unhappiness. At least until you tell it to build a Courthouse.
If you've only taken the one city and your happiness isn't in the toilet you may as well just annex it.
Other than that Iron Curtain exception, there is no downside to puppeting first and then maybe annexing later. (my problem is I often never annex puppet cities long after they could actually be productive.)
You take a bigger happiness hit with immediate annexation. That means slower progress toward your next golden age and it might put you into unhappiness, always an ugh for several reasons.
Annex or puppet, nothing can be built by you for a certain number of turns, quite a few if it is a large city (number of no-action turns is shown with the red fist in the city marquee when you zoom in on it), Nor can you override the city manager to direct the workers for that period.
I like to have a fund for buying a courthouse as soon as the no-action (resistance) turns end popping happiness back up ASAP in that way.
Be careful when you get offered a bunch of cities in a peace dea, it may make you go negative happiness.
I totally miss this in Civ VI.
Choosing to puppet a conquered city defers these costs until you later decide to annex it.
Sometimes I annex a city upon capture, or later on, if I want to build units in it.
You may be building or are about to build something that requires all of your cities to have a building that the city you captured does not. For example, Building Oxford University first requires universities in every city. If you puppet a city that doesn't have a university, you can keep right on building Oxford, finish it and then annex. You can then build a university (or not) in that annexed city as it is no longer needed for your already built Oxford.
But if you annex right away then Oxford is halted until you build (or buy) the university in the annexed city and you can't even start building that university you need until resistance has ended in that city.