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Wow, your question got me wondering so I took a fast look and it appears that magnet wire and enamelled copper wire are generic names for a rather broad range of products. Looks like there are a million choices for enamels and as many for the wire itself.
I believe that you may find that one of the differences between the two is that the magnet wire is soft-drawn copper and will better retain its shape when it is bent, making it nice for coil and transformer winding in that it won't as easily spring open when tension is released. The heavier gauges are also easier to bend around coil and tranformer forms.