In The Godfather, the somber waltz on solo trumpet appears as a leitmotif whenever there is a suggestion of business taking place
Juxtaposing Nino Rota and Wagner is audacious and interesting. They both had a profound impact in popular culture through their music. Rota's music is still haunting in so many ways. Normally, I would never have thought of this. But it is also a way to look at popular music as the inverse of classical music and how they both can reach our souls.
I would say the composer of The Godfather shamelessly lifted a theme from Donizetti’s Don Pasquale as explained here: http://www.good-music-guide.com/community/index.php?topic=17536.0
If you want to read more, these are two great articles on Wagner (and his music’s tense history):