Copenhagen is an interpretation where I have one mind, you have one mind, and each of us have one thread of experience. There are numerous places along that thread where the physics to calculate the time evolution of that thread is not deterministic, where a random choice has been made.

MWI is an interpretation where I have many minds, as opposed to the one mind I have in Copenhagen. In the MWI interpretation, each of my minds exists in a separate and non-interacting universe from all the other versions of my mind. If I wonder as I type this why this version of me is the one in THIS branch, MWI has no theory for that. MWI tries to make that question seem less interesting by pointing out that there are lots of versions of me asking that same question, so somehow obscuring the me-ness of the me in this branch with the me-ness of all these other similar but not identical me's in these other branches would render the question meaningless.

But as an interpretation with no observable experimental differences, MWI and Copenhagen are likely to have the same number of random events dictating progress. In MWI, the randomness is isolated to just one of many me's which of course is still quite unique and interesting to me, but which is not as bad as Copenhagen where it is the entire universe that got changed by each random waveform collapse.