Knowing the film version pretty well, I had to distinguish mere familiarity from aesthetic preference. But after many listens, I think the 1964 version comes off better overall.
The main reason is that Rex Harrison's phrasing is much, much superior in the film version. The lack of meaningful emphasis in his 1956 songs is remarkable in contrast to the 1964 album. Possibly he'd had longer to think about them and develop how to enunciate. He may also have done better speaking the lines in context rather than in studio:
Rex Harrison declined to pre-record his musical numbers for the film, explaining that he had never talked his way through the songs the same way twice and thus couldn't convincingly lip-sync to a playback during filming (as musical stars had been doing in Hollywood since the dawn of talking pictures). In order to permit Harrison to sing his songs live during filming, the Warner Bros. Studio Sound Department, under the direction of George Groves, implanted a wireless microphone in Harrison's neckties, marking the first time in film history that one was used to record sound during filming.Andrews is of course delightful, but I would even venture the heresy that Nixon is a more convincing Cockney than Andrews.
Some of the minor characters are difficult to judge by the albums, but Robert Coote's Pickering sounds inferior to 1964's Wilfrid Hyde-White. Thank god they got Stanley Holloway for the movie too!
Also, some of the lyrics are better in the film. It's a shame to listen to the 1956 version and not hear the magnificent rhyme,
"I know each language on the map," said he,Cole Porter should've been jealous.
"And she's Hungarian as the first Hungarian Rhapsody!"