Poetic theme is the main point the author is trying to make with the poem. Theme is the lesson that readers learn about life after reading a poem. Another way to think of theme is as the "moral" of the poem. Many poets have a thematic focus on major life issues such as love, death or independence.
The subject of a poem is the topic, or what the poem is literally about. Poets can write on any topic imaginable, as long as they make it appropriate for their audience. Common subjects for poetry include nature, growing up, growing old, children, and life events. Because the range of possible subjects is so broad, writers like Shel Silverstein can write directly to children, while more esoteric poets like William Wordsworth can cater to adults.
Example: "Birches" by Robert Frost
On the literal level, the subject of "Birches" is a man looking at bent-over birches, wondering if a snow storm has bent them or if a young boy was swinging on the birch trees because he was too far away from town to play with friends. The subject is like the topic of the poem, or what it is about. The theme goes much deeper. The poem insinuates that swinging up and swinging down are reciprocal aspects of reality, like heaven and earth. The poem's theme also expresses the desire to stay on earth a while, rather than the to succumb to life's reciprocal element, death. One of the last lines is, "Earth's the right place for love." This also suggests a theme of loving on earth during our short time here.
Example: "I Hear America Singing" by Walt Whitman
This poem is a litany of workers. The subject of this poem is the unique songs that each of the many types of workers in America have to sing. Whitman writes about "The carpenter singing his as he measures his plank or beam, / The mason singing his as he makes ready for work, or leaves off work." At first glance, readers might think this poem is simply a list. The theme of the poem, however, speaks to the way the many American workers join together to form a well-oiled machine. In this poem, the workers in American are greater than the sum of their parts -- a strikingly positive theme for Whitman.