4

Наконец я вас догнал.

vs: Наконец я вас настиг.

These two verbs essentially mean "catch up with someone", but I wonder if and how they are nuanced?

3

настичь is purely bookish nowadays, using it in mundane conversations makes one sound altisonant

however unlike догнать it also can mean to catch up unintentionally, without chasing after someone, and in this meaning it's synonymous with a more colloquial нагнать

for denotation of catching up in figurative sense (catch up to, overtake in knowledge, physical features, accomplishments) догнать is currently the only appropriate verb

the difference in register between them reflects in the way i subjectively perceive your examples: in the 1st one вас rings you plural (German sie), in the 2nd it rings you singular polite (German Sie)

2

"Догнать" is a neutral term and can be used in most situations. "Настичь" is mostly applicable to the forces of nature - "Нас настигла буря" is still more common than "Нас догнала буря".

"Наконец я вас настиг" sounds like what a villain would say in a comedy (or perhaps anyone with a sense or irony).

2
  1. Наконец я вас настиг.

    This is absolutely bookish, looks like a citation from a book written in 19th century.

  2. Наконец я вас догнал.

    This phrase looks like an end of the story, which you tell to a person, who you caught up:

    [A very-very long story about "a chase" ...] ... Наконец я вас догнал.

  3. If you want to say this phrase to someone right in the moment when you have caught him/her up, it is better to say:

    Наконец-то я вас догнал.

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