sich setzen is to sit down
sitzen is to sit / to be sitting

the first is an action, a movement, the second is a state of being, a position.

I like to sit by the window / he sat down by the window - Ich sitze gern am Fenster / er setzte sich ans Fenster.

is that clearer?

IMHO
sich setzen = to take a seat, to sit down
sitzen = to sit (on a chair etc.)

Erst "setzt man sich", danach "sitzt man". "Sitzen" ist das Ergebnis von "sich setzen".

"setzen" is "to sit down" --> the act of placing yourself on the chair or floor or whatever other sitting surface.

"sitzen" is "to be sitting" --> what your do on the chair once you have sat down until you get up.

To me (German native) there is not much difference in formality, but "Darf ich mich setzen" would be the usual way of putting it - simply because you are asking to place yourself on that chair; you are not asking whether you could stay once you have sat down.

On a train I guess you could say "Kann ich mich hier hinsetzen?", though "Ist dieser Platz frei?" would be more common.

"Sitzen" is only used as an active verb when talking to a dog, isn't it? Sitz!

(And can anyone tell me if I am imagining it, or if there is some old comedy sketch based around the phrase "Setzen Sie sich doch"?)

It's a bit clearer now, thanks. So I can't really say "kann ich hier sitzen" in that case, unless as 3 said I'm asking for permission to stay there afterwards, right?

#5 You can, though you might use it to mean "can I keep on sitting here?" - you're asking if the state of you sitting on that seat is OK, rather than the act of sitting down.

So I could sit down unannounced and then say "ich darf doch hier sitzen oder?" or something then....but to ask in advance it would have to be "darf ich mich setzen"...complicated but I understand

Please note that "sich setzen" is connected with an accusative object, asking: "where to?" and "sitzen" is allied with a dative object, introduced by the question "where - in what location?"

I think you are making it too complicated. Isn't it more or less the same thing in English? Whenever you can use "sit down" or "take a seat" in English, use "setzen".

"Bitte, setzen Sie sich doch". (Please, take a seat)
"Mir ist schwindelig. Ich muss mich setzen." (I'm feeling dizzy. I have to sit down).
"Er setzte sich unbeholfen auf den Stuhl." (He sat down awkwardly on the chair".

Otherwise, use "sitzen".

Er saß nun schon seit einer Stunde auf dem Sofa, und seine Frau war immer noch nicht gekommen. ("He had been sitting on the couch for an hour, but his wife still hadn't arrived.")

In the "train example", would you really ask in English "May I sit/sit down?" As the others have already mentioned, in German at least you would probably say something along the lines of "Ist dieser Platz noch frei?", so neither use "sitzen" nor "setzen".

#9 In English you would say "Can I sit here?". "Sit" can mean both "sich hinsetzen" and "sitzen"; that's why Queen wanted to know the difference. Siehe Wörterbuch: sit "I sat on the chair" can mich "Ich saß auf dem Stuhl" or "ich setzte mich auf den Stuhl". You only need to add "down" when the context is not clear.

I would say "can I sit here (please)" every time (NOT with "down"!)

mich --> mean (brain going into meltdown...)

Okay, I rephrase: When you have an English sentence like "Sit, please", determine whether you can substitute the verb with "sit down" or "take a seat". When the meaning remains unchanged after the substitution, you must use "setzen" in German.

If the "substitution test" doesn't work, you must use "sitzen" in German.

P.S. Have mercy on my feeble attempt at explaining German grammar...;-)

In the "train example", would you really ask in English "May I sit/sit down?"

I would probably ask "is anyone sitting here?"

#13 oh, I see what you mean.

#7: Genau. Die Frage "Darf ich hier sitzen?" drückt streng genommen aus: "Darf ich mich hier sitzend aufenthalten?"

Es ist aber mit sitzen und setzen nicht komplizierter als bei anderen, ähnlichen Wortpaaren wie z. B. stehen/stellen oder liegen/legen. Speziell bei liegen/legen gibt es ja mit to lie/to lay sogar eine englische Entsprechung, würde ich sagen.

Schwierigkeiten haben allerdings auch viele Muttersprachler mit dem analogen Paar hängen (hing, gehangen)/hängen (hängte, gehängt), weil die Präsensformen identisch sind.

English people often have trouble understanding the difference between "lie" and "lay", in their own language. Perhaps as other verbs, like stand and sit, are so multi-purpose, plus the fact that we tend to use "put" for everything, whatever shape it is or position it's put down in. We've had quite a few threads asking about these verbs.

@17 - I know, and it irritates me no end! I DO understand the difference, so now I can even better understand sitzen vs. sich setzen :)

@8:
mit Verlaub, "Sitzen" zieht doch kein Dativobjekt nach sich! Es ist ein intransitives Verb! Die Frage nach dem "wo?" ist doch keine Frage nach dem Dativobjekt! Da verwechselst du die grammatikalischen Begriffe.
"Setzen" fragt zwar, wie du richtig behauptest, nach einem Akkusativobjekt, aber nicht durch die Frage "wohin?", sondern durch die Frage "wen?"