When writing an analysis of poetry, one should talk about it in terms of its technical aspects as well as its content and the historical/cultural context in which it was written. If you are going to speak ONLY about the "types" of poetry, as you said in your question, then you can focus purely on the technical features of both types (for example, how are a sonnet and a piece of lyric poetry similar and/or different from one another).
On the other hand, if you already have two specific poems in mind to use in your compare and contrast paper, then it would be a good idea to try to find as much information as you can about them so that you can see which elements are the same and different between them.
Here is a list of some important details you should try to identify about each of the poems you have selected, which you can then organize into paragraphs in whatever way makes the most sense for your particular assignment:
- Author's biographical information (especially experiences that influenced his or her writing)
- Historical era or period in which each poem was created
- Summary of poem's content including what it is about, who its speaker is, the speaker's attitude toward the subject, etc.
- What type of poem is each selection? Examples might include Shakespearean sonnet, Petrarchan sonnet, limerick, haiku, epic, elegy, lyric, ballad, free verse, concrete, and so forth.
- What is the actual structure of each poem selected? How many lines does each have, and how many syllables are in each line? Is there a rhyme pattern you can detect? If so, what is the pattern? Is there a regular rhythm you can detect? If so, what type of rhythm units are being used (iambic, trochaic, spondaic, anapestic, dactylic)? How many "feet" of those rhythm units are found in each line? How many stanzas does the poem contain? Does the poet use conventions such as punctuation (or lack of it) in an unusual way or not? If so, is there a reason the poet used it in that way?
- What choices has the poet made in language and style? For example, is there a specific diction, symbolism, figurative language or otherwise interesting word choice that has caught your attention?
Once you have answered these questions for each of your poems, you should then be able to talk analytically about whether they have things in common or differences between them. You will be able to make statements such as "Poem A contains three stanzas that each have four lines with five iambs in each line, while Poem B contains four stanzas that vary in their number of lines and feet. But both poems follow a rhyme scheme of ABCB repeated throughout them" or "Although this poet is best known for his short stories, he has written poetry in a variety of forms as well. His ballad 'Poem A' has similar themes as his lyric 'Poem B', even though they are different in structure and length".