What are the similarities and differences between lyric and epic poetry?

The main similarity between lyric and epic poetry is that they're both primarily concerned with telling stories of one kind or another. Lyric poetry is poetry of the emotions; more often than not, a poetry of love. As such, it often has a story to tell, whether it's of the speaker's emotions or of his quest to find love.
It's the latter aspect of lyric poetry with which epic poetry has the most in common. Epic poems are concerned with telling stories, but they are more than just subjective expressions of feeling. They deal with larger than life characters, both mortal and immortal, whose extraordinary adventures give shape and meaning to the world in which we live.
Epics, like lyric poems, were originally designed to be recited to the accompaniment of a lyre—which is where we get the word "lyric" from. In many cases, they weren't written down; they were passed down from generation to generation through the medium of song.
In lyric poetry, the main emphasis is on the individual. In epic poetry, it's on humankind as a whole. Lyric poetry deals with the specific, the concrete, whether it's an emotion or a philosophical insight as in Wordsworth. Epic poetry, on the other hand, deals with the universal, with general ideas and situations that speak to humankind in general by providing objective truths.