Translation:The police officers were chasing the young criminal in the theatre.
I think "kerget" is more literal and more specific. And closer. You run, I run after you, you are in my sight, that is "kerget".
"Üldöz" is a synonym but it is not necessarily a physical race. It can mean "fight" or "be against" something. For example, "crime fighters" are "bűnüldözők".
It might be confusing that the official translations given here and for the sentence with "üldöz" (https://forum.duolingo.com/comment/21258644) use a different verb tense ("chased" vs. "were chasing"). I asked a native speaker, and according to him there is no such difference in meaning, only the difference indicated by vvsey. "Kerget" also sounds slightly more playful to him: "A kutya kergeti a farkát" vs. "Az Illuminati üldöz engem".