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The basic advantage is that a core-xy has less moving mass than other designs, which translates to higher acceleration and speeds, with less ringing. Also any movement the printhead makes uses the torque from both the A & B motors no matter what direction it is moving.
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Yes, except the ringing. Unless something has changed, Gantry and CoreXY both actually tend to have worse ringing than Cartesian designs.
The biggest advantage over cartesion is your not moving the mass of the part you are printing, except in the Z. This means you don't have to worry the leverage that it generates. Essentially, you can print tall skinny things much easier, same as the old gantry printers.
Disadvantages. They cost more than a Cartesian. The low mass leads to them more often being bowden hotends. They are suppose to be a pain to troubleshoot if you start having dimensional issues in the xy since both motors control both directions of travel. Also there are more moving components in the corexy than a gantry or Cartesian system so that is more parts to troubleshoot/replace.