Since these chords aren't practical for string quartet, I've retained four-part writing throughout in my transcription:
In his string arrangement, Beethoven uses a different technique to support the crescendo: He gradually increases the spacing. The parts begin in the same octave and finish with the bass and soprano more than three octaves apart.
In phrase one, Beethoven raises both the bass and tenor one octave, placing the viola above the second violin. In the second measure, this close spacing crowds the viola out. There is no good note for the viola to play on the first beat of this measure. A D, as in the piano version, would anticipate the resolution of the melody (which will sound worse in strings than it does on the piano). Doubling the cello's F would be fine at the octave (as in my transcription). But a unison doubling on a note that screams for a strong vibrato is not a good idea. So Beethoven omits the viola part altogether and switches to a three-voice texture.
On the cadence, Beethoven has the cello drop down an octave for the second G. Repeating the high G would not work, since the final chord would not be in root position. This octave drop supplies Beethoven with a new motive, which he duly exploits. He gives the cello another octave drop at the end of phrase two and an octave rise at the end of phase three. This octave rise turns the last two notes of this passage into an echo. Very clever. Beethoven takes an idea born of necessity and turns into a unifying feature.
The final change Beethoven makes to phrase one is to add a sforzando. Perhaps he thought the six-voice chord gave the climax sufficient weight in the original. In the string version, the number of voices is actually reduced in the climax, so the sforzando becomes necessary.
Phrase two returns to four-voice writing, with the bass and tenor again raised an octave. This time, however, the second violin takes the raised tenor part (starting on beat three) and the viola takes the alto. Why didn't Beethoven do this is the first phrase as well? Apparently he prefers to give the second violin whichever part moves in similar motion to the first violin. In the first phrase, that's the alto part; in the second phrase, it's the tenor.
In phrase three, both the tenor and bass drop down to their normal register. (The new octave-drop motive now serves yet another function: preparing this register change.) The second violin doubles the melody an octave below, leaving the viola to cover both inner voices with double stops. The combination of the octave doubling of the melody and the lowered bass effects the grandeur supplied by the chords in the piano version.
At the cadence, Beethoven adds a retardation in the viola part so that it moves in thirds with the melody.This is one of those changes that are obvious in retrospect. Beethoven is simply paying more attention to good voice leading in the string version than he did in the piano version. Note the crisp effect of having staccato in the bass while the other three voices play legato.
The final measure puzzled me at first. Beethoven chose to fill out the final chord by adding a D. But why give the D to the second violin? Beethoven went out of his way to cross the parts, so there must be a reason. The reason, presumably, is that the B is the "melody." But so what? We are going to hear the higher note as the melody no matter which violin plays it. Then it occurred to me. Yes, we will hear the D as the melody. But Beethoven wasn't writing for us. He produced this arrangement for amateur string players who get together on Sunday afternoons to play through some popular pieces. The performers themselves will be aware that the B is being played by the first violin. So I suspect they will hear the B as the melody.
At least that's my guess. Another possibility is that the D was an afterthought. Perhaps originally the second violin was silent on this chord. Then Beethoven decided to add a D. Rather then scratch out the first violin part and rewrite it, he just gave the D to the second violin. As someone who learned to compose before music notation software and actually worked with dip pens and onion skin, I can attest to the temptation to do such a thing. But, if you look at some of Beethoven's manuscripts, you can see he wasn't shy about crossing things out. So I'm not sure that explanation holds up.
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