Learn Russian: Началась метель, так что наш рейс задержали. - A blizzard began, so our flight was delayed.

Both words express a consequence, but grammatically they are different. Поэтому (therefore) is an adverb and often starts a completely new independent sentence. Так что (so / with the result that) is a conjunction specifically used to attach a subordinate result clause to a main clause. In casual speech they are often interchangeable, but так что weaves the cause and effect into a single, smooth thought. Note that the comma always goes before the whole phrase так что, never splitting it.

Why does the verb началась come before the subject метель?

In Russian, placing the verb before the subject is the standard way to introduce a new event or state of nature into the scene. It translates functionally to "There began a blizzard." If you said Метель началась, it would sound like you are giving an update on a specific blizzard the listener already knows about, similar to "The blizzard began."

The translation says "was delayed", but isn't задержали an active verb meaning "they delayed"?

Yes, it literally means "they delayed." Using a third-person plural verb without the pronoun они (they) is the most natural way to express a passive action in Russian when it doesn't matter exactly who did it (in this case, the airport authorities). While you could use a true passive like рейс был задержан, this indefinite-personal construction is much more common in spoken Russian.

Is наш рейс the subject of the second clause?