The brushes above are
So what's the difference between a quill and mop brush?
A quill is a main wing feather or tail feather of a bird. Shown above is the quill of a parrot. Traditional quill brushes were made in the past by using the quill the hold the hair together, and tighten with wire or string.
With modern quill brushes, plastic (often transparent) is used to hold the hair together, and tighten with wire.
A quill brush differs from other brushes by how it's made. A quill brush may or may not be a mop brush. E.g. A small quill brush is probably not used as a mop brush.
A mop brush is a brush designed to hold plenty of water for the purpose of painting over large areas easily. Most mop brushes are made of either real or synthetic squirrel hair. And there's almost no significant difference between the performance of a real or synthetic squirrel hair brush.
Quill brushes may hold a sharper point compared to squirrel brushes. But that really depends on the hair that's used. Because mop brushes use squirrel hair, they are typically floppy to handle, just like a real mop.
Mop brushes do not need a sharp point because they are not meant to paint details.
One thing common about mop and quill is, it seems that the ferrule is quite short, at least shorter compared to the long metal ferrules of round brushes.