This is the present perfect tense - "he wrote" is the past tense. If he has written a beautiful letter, then the letter remains written right now and it remains beautiful. He WROTE a beautiful letter suggests that the whole event surrounding his writing of it has passed. In this particularly example, I would say that it's like "Look Jane, he has written a beautiful letter, you must read it!" whereas "When my husband and I were young, he wrote a beautiful letter to me on my birthday."
in the latter example, it would sound weird (and in fact would be incorrect) to say "When we were young, he has written a beautiful letter."