Ask HN: What's the difference between the "new iPad" & the "iPad with Retina"?

What is the difference between the "new iPad" (iPad 3) and the "iPad with Retina display" (iPad 4 - that went on sale today)?

I thought the 3rd generation one had retina? What are the main differences between the two? I'm looking at getting one but slightly confused. Also, if I were to get a case, would a case for the "iPad 3", work for the "iPad 4"?