Usually, if 'dress' is used to describe an item of clothing, it means that it is used for a formal or important occasion. Here is a journal from 1888 describing a 'dress bonnet' among other fashions of the day "A very charming full dress bonnet is made of velvet put over the frame full, and over this covering are passed three embroidered bands, one on top, from front of brim to crown back edge, and the others across sides".
https://upload.wikimedia.org/wikipedia/commons/1/14/Ladies'_...This next item says of dress bonnets, "This type of headwear would be worn by the middle and upper classes. Dress bonnets are generally made from silk or straw and can also be sheer. Decorations include fabric trims, lace, ribbon, silk flowers, feathers, grasses and tassels. A dress bonnet is not intended as protection for the sun, weather, or the cold. The shape of dress bonnets changed over the war years."
http://www.txcwcivilian.org/id111.htmlSo, I would assume that a 'bonnet' was a plain, everyday bonnet, and a 'dress bonnet' was an upgraded version that was much fancier. NB bonnets are different from hats - see photos