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Hi!
Both poem and lyric can have the same meaning, but the difference between poem and lyric is that lyric is usually used when we are talking about words of songs, poems have a rhyme, lyric doesn't have strict rules it can have rhyme or not. That is how I understand it.
To write poem is very hard. First you need inspiration, that to know several rules to make rhyme and nice words with deep meaning. I mean you have to know how to use simple words to make deep sense. Good luck!
what is poem? 
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