Can anyone explain the significance of the small notes in the following music? An otherwise identical arrangement in a different "easy" song book shows only quarter notes without the smaller notes. What is the difference in how it's played? And is there a name for what these small notes represent here?
Despite my comment, I'm going to go out on a limb here and guess that this is a strophic song, with verses having different numbers of syllables. The small notes are to accommodate the extra syllables in those verses that have more of them.
When I can think of an example, I'll edit this answer to add it.
As already mentioned, this is a way to notate a strophic song. Depending on the type of song, it could also be described as having an "irregular metre".
This is described on Wikipedia here https://en.wikipedia.org/wiki/Metre_(hymn)#Terminology_and_abbreviations
A few hymns have an inconsistent metrical pattern across their verses. Such a metre is described as '"irregular"; one well-known example is O Come, All Ye Faithful.