What are the differences between song lyrics and poetry, poetry and rap?

The short answer to this is that both song lyrics are rap are, in fact, poetry.
The term "poetry" has both a more literal, technical meaning, and a deeper, somewhat figurative sense in which it is used. Most people conventionally think of poetry as a form of writing in which there are both rhyme and a regular metrical pattern, written out line by line (the lines known as verses) and with the lines often grouped into larger units known as stanzas. This is the first, most restricted definition. But a huge amount of literature exists which does not follow this definition, and yet very few knowledgeable people would deny it is poetry. Walt Whitman's poems, for example, for the most part have neither meter nor rhyme. This is called "free verse." Whitman was a pathbreaker, yet, traditionally, some of the greatest English poetry, such as Shakespeare's and Marlowe's plays and Milton's Paradise Lost, has been written in "blank verse," which is metrical but doesn't rhyme. We also ascribe the term poetry to religious writings: for example, the Old and New Testaments, the Quran, and the Hindu Scriptures, which are written in verse form. And in the broadest sense, even prose works can be labeled poetic; one can speak of Fitzgeralds's novels such as The Great Gatsby as being "sheer poetry." The answer to this broader definition lies in the nature of the words themselves that form a piece of writing. What we define as "poetic" is strongly imaginative, rich, unusual in word choice, and emotionally moving. Often prose, like that of Fitzgerald or Joyce, has a musical quality and sounds pleasing to the ear in ways that most ordinary language does not. And language that is metaphorical—expressing ideas not literally but having one thing represent another, drawing comparisons between unlike ideas and evoking an especially emotional response—is poetic regardless of the outward form in which it is written.
Given these definitions, both popular song lyrics and rap are poetry. The elements of rhyme and meter are generally present in both forms. The presence of music, which accompanies the rapper and to which the words are sung directly as in the popular song—enhances the emotional mood of the words. In classical music, composers such as Schubert, Schumann, and Brahms set to music poems which had already been written independently are complete works of art on their own. In the popular song today, the lyrics and music are usually written as part of the same project, but the example of the classical composers shows that the addition of music does not somehow prevent the lyrics from being poetry in their own right. So, finally, in answer to your question, there is no objective difference between "poetry" and the lyrics of contemporary songs or of rap.