What mystery you are talking about? When you measure a particle and learn something about it, we erase the previous wave function, and replace it with a wave function that corresponds to what we found.
Just like if I calculated the odds of you being dealt a royal flush of diamonds in 5 card stud (1/2598960). But if I then peeked at 4 of your cards and find the jack, queen, king, and ace of diamonds, that will cause me to erase the previous probability, and replace it with a much larger probability (1/48). It isn’t a mystery that gathering additional information about an object, can change probabilities of outcomes associated with the object. This isn’t magic.
the same as what?
What random measurement are you talking about? No one has mentioned random measurements up to this point (as far as I can tell).
A difference between what two things? I have no idea what you are talking about at this point; maybe you could start over and describe EXACTLY what exact situation/experiment you are talking about?