How does lyric poetry differ from narrative?

Narrative poetry tells a story. Narrative poetry has no obvious 'point of view' - since what is happening (the plot and the characters, and sometime the setting) is what the poem 'is about'.

(We can often discover a 'point of view' in a narrative poem - but we need to look for it (in fact this is the business of advanced literary criticism)).

Lyric poetry originates as song. ('Lyric' means 'sung with a lyre accompaniment': early poets accompanied themselves on the lyre, much as modern poets often accompany themselves on a guitar). In lyric poetry the poet talks about something he has seen, heard, felt, or thought. There is an obvious 'point of view' - the point of view of the poet.

Robert Southey's 'Bishop Hatto' is really a narrative poem. There is a meaning there, but it is told through a story.

John Clare's 'I am' is a lyric poem. It has a story, but the story is told through the meaning.

It is much easier to learn how to distinguish a lyric from a narrative poem than it is to precisely define what the difference is.