Copenhagen Interpretation and MWI don't make different predictions from each other, but there are other theories that do. For example, pilot-wave theory makes different predictions. There are also some superdeterminist theories that make new predictions (Gerard T'Hooft is one advocate of such theories, I believe). Unfortunately I don't know of any concrete examples. Here is some more information on the subject from Sabine Hossenfelder:

http://backreaction.blogspot.com/2019/07/the-forgotten-solut...