A poem is more like a song that's written in words and that rhymes or at least rhymes in a certain kind of way. It is a literary piece written in verse
These days poems that constantly rhyme are frowned upon as child like a poem should touch the heart of the reader, also be meaningful.
Where as a limerick is a funny little poem containing five lines. It has a very distinctive rhythm and rhyme pattern., witty and humorous, also know as nonsense poem, made popular in it's present form by Edward Lear
A rhyme is a repetition of similar sounds in two or more words, most commonly associated with children's nursery rhymes, repeated in parrot fashion until the rhyme is know of by heart.
Many nursery rhymes, have some historical figure who the rhyme was based on
I wrote this as my wife is keen on writing poetry, but seem to thing they have to rhyme.