When you save in Picard, it changes the Artist, title, etc., but it also adds MusicBrainz custom tags to identify the Recording, Release, Artist, etc. When you load a file back into Picard, it checks those MBID’s and assigns all the data for those entities back to the tags for the file. I’m assuming that when you say you changed it manually, you went in to the file and typed in artist name, title and so on.
The more persistent way to fix something manually when you have a file assigned to the wrong recording in Picard is to find the correct release on the website, drag the website address bar to Picard, so it opens up in the right pane, then drag the file in Picard to the correct track on that release. When you hit save now, it will save the correct visible tags and the correct MBID’s, so next time you pull up the file in Picard, it will be matched to the correct recording.