"His speech was both interesting and important."
Translation:Hans tal var både intressant och viktigt.
"både ... och" is the conjunction form meaning "both ... and." Another example would be "Han talar både tyska och engelska" meaning "He speaks both German and English." The phrase doesn't have to be separated, as the answer to the question "Vill du ha kött eller grönsaker?" (Do you want mean or vegetables?) could be answered "Både och." (Both). This is a shortening of the entire sentence "Jag vill ha både kött och grönsaker."
"båda" on its own is a pronoun, meaning it would substitute for a noun or noun phrase. "Skriv på båda sidorna av pappret" = "Write on both sides of the paper." "De båda ankorna simmade" = "Both of the ducks were swimming."
I'm not a native speaker but that is my understanding!
It's my understanding that sitt, sin, and sina all have to refer back to something. So if this sentence were part of a longer discussion where you'd been talking about han, then sitt could potentially work. But in this case, where all we have is a sentence by itself, there is no han already present for the sitt to be talking about. If that makes any sense.