There's almost nothing between the songs (or during them), aside from little incidentals and snippets of one-sided conversations. During "See I'm Smiling", for example, Cathy is continually talking on the phone, trying to get hold of Jamie. There is also one "monologue" for Jamie, which is essentially him doing a reading from one of his own books. None of the text provides significant additonal information, IMHO.
I put monologue in quotes above because that passage, like all the text in the show, is purely incidental and does not drive the development of the characters (in my opinion a true monologue should involve some change or development in a character, or it's pointless). That all happens through song in this show... and what you get on the CD is basically the raw material for the show itself.
Interestingly, the stage directions in the script are rather "pregnant" (i.e. they imply a lot of activity/expostion that is not otherwise expressed in song or text), leaving a great deal of work to be done with creative staging.
I have found the full libretto to be somewhat helpful for imagining how the music can serve as the source for a full production, but most of that information is, as I've noted, in the stage directions.
"There are only two tragedies in life: one is not getting what one wants, and the other is getting it." -Oscar Wilde
