Wrote is the simple past and stands alone: "she wrote the report." Written is the past participle and needs a helper: "she has written the report", "the report was written." The test is the same one that separates took from taken — look for the helper verb.
The core difference
Both are past forms of write, distinguished by whether a helper verb is present.
- wrote — no helper: He wrote back the same day.
- written — with have/has/had: He has written back.
- written — passive, with be: The letter was written in haste.
Written as an adjective
Written also works as a plain adjective where wrote never can: a written agreement, written English, put it in writing. That use is unrelated to the helper-verb rule and is another reason the two are not interchangeable.
The test
Look for have, has, had, was or were. If one is there, use written.