A sweater with an open front fastened by buttons or a zipper is generally called a cardigan, but the nomenclature for other styles in different dialects can be quite confusing. In British English, a sweater may also be called a pullover, jumper, or jersey.

What is it called when a sweater has a zipper?

Hoodies with zippers are generally referred to as zip-up hoodies, while a hoodie without a zipper may be described as a pullover hoodie.

What do you call a hoodie without a zipper?

Jumper will always have a headgear. Jumper is just another name for a hoodie without a zipper. There is not a word of non-conforming jemper, if any, it's another name for a garment called a sweater.