The words do have the same meaning, but most people would choose one syllable rather than say five. Modification just sounds more serious to me. I might make a slight change to a report. For a moon rocket or a brain surgery procedure, I would make a modification.

"...made a slight change (modification)." [Need the article 'a'] They are basically the same. One is not better than the other. Just don't "do a re-write".