It's a darker, sadder variant of the same, basically F major harmony. The bass note is still a long F. The falling internal melody is given some chromatic decoration in the second version.
It's an interesting song - I could probably get an entire theory class out of it! The variant harmony, with its potential to be construed as an F7 chord, could have been used to lead into the middle section in B♭ major - 'through the summer and the fall' - but it isn't. It's used just once leading into a Dm chord - 'so the years go by'. The E♭ isn't functional, just a touch of colour. Do we let the D♭ double up a C♯, leading note of D minor? It doesn't really ACT like a leading note, does it? But good writing is full of this sort of subtle 'maybe' that makes THAT note the right one in THAT place.
Happy songs are in major keys, sad songs in minor keys - right? Well, not this one. A pretty solid F major.
Though he often prefers to compose miniatures, Randy Newman has a very solid musical background. Look up his family tree! I wonder if he consciously borrowed from the Edith Piaf song 'Hymn of love':
The movie version is actually in G♭ major, a semitone up from your printed copy.