How does writing it down make any difference, unless there was a corroborating witness during the conversation? It seems like it would still be he said she said. Is it just a case of writing it down being more unlikely for the memory of the words to be distorted by time passing?
For some weird reason, keeping a journal of stuff as its happening (eg your neighbours making noises) seems to hold up better legally than just stating these things verbally.
One theory: I guess because the journal commits you to one story. One consistent story is easy to shoot down, eg your neighbour claims and proves they were on holiday when your journal speaks of lots of noise.
Evidence that is easy to shoot down, but doesn't actually get shot down, is more believable afterwards.
When you go report something to HR, the HR rep always writes things down on paper. Why do they do that? In case things go to court.
Write down on paper what everyone says. Otherwise is your word vs the manager's. Employees will not win in a case of he said she said.