The old legacy export saves an anti-aliased (8bit) alpha channel to mask the object, while the new (arguably broken) export exports an aliased one (1bit).

One additional crucial difference between the two is that the legacy export uses a black background, while the new one uses white.
It can be fixed, but you will have to open the file in an image editor, decouple the mask, remove the white or black by using an unmultiply type method (which is not available as a function in Photoshop, unfortunately), and create a new mask based on the newly created transparency.
Ideally the new export would generate an anti-aliased alpha. It is a bug, and known issue. The new export is useless because of this. Hopefully this will be fixed. For now, use the old export.
Export to PNG problem