[link to original on tumblr]
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I looked up the Moonlight Sonata on Wikipedia, and I learned how to play (rather poorly) the first three measures:
The octaves in "Moonlight Sonata" descend diatonically (C#, B, A) before going to F# and then ascending to G#. The octaves in "From the Sun to the World" are much simpler. For the first two chords, it's A. Then it drops to E before returning to A.
The first three measures of the Moonlight Sonata arpeggiate C# minor, A major, and D major, and the arpeggiated chords in "From the Sun to the World" don't really resemble these at all. I'm fairly certain that it's A minor, D minor, E major, A minor.
So, as far as the parts, there's some resemblance, but there isn't much in common note-wise.