Poet's generally write and recite, whereas bards write, recite AND sing. As well, poets are more general in their topics, while bards focus on legends, myths or a more narrow theme. That's my interpretation, anyway.

'Bard' is just an old-fashioned, and/or romantic word for poet. I don't think that anyone would seriously use the word 'bard' for a modern poet.

Greg's exactly right. Being a "bard" was a traditional of oral storytelling in the 'old days', and often bards were paid by a patron such as a king or noble to sing poems or stories about the heroic deeds of that person. Shakespeare is often described as a "bard" because he wrote rhyming verse for the stage. In contrast, a "poet" usually (not always) uses the written form. There is such a thing as "performance poetry" where someone stands up and tells/recites their poem in a dramatic way, and that's probably the closest thing we have in 2019 to a person being a "bard".