I've been listening to this song a lot lately, and Billy Joel has always been a favorite anyway- He gives a lot of emotional intensity to this song
Each of the verses are strophic. The chord structure for the verses is pretty straightfoward: V7-I/V7-I/I-IV/vi IV/ (first period) V7 I/V7 III/vi/IV V7 I (second period-very clear pac- it's tonally closed. The bass line is very noticable, and gives the running 16th notes in the right hand and in the guitar a very grounded, more fluid motion, pushing it forward.
The refrain moves through this ii-V-I progression, and the time seems to move from 2/4 to 4/4. the bass ostinato is still extremely audible, with the constant 16th notes over it.
This song feels like it could also be in 3/8-it's almost like a waltz.