Honey

By the crude definition of vegan, honey isn't vegan because it's an animal product. However, in spirit this is complicated somewhat by the legitimate and necessary use of bees to pollinate many crops (see Bees in Agriculture), and while most commercial operations over-harvest honey and replace it with sugar or corn syrup, honey can potentially be a legitimate byproduct of strictly pollination-focused operations (see here).

Nutrition

Honey is not a healthy food product. Nutritionally, it is essentially sugar.

The difference is that it's often contaminated with potentially dangerous bacterial spores which can cause a botulism infectinon, and is not safe for young children and the immunocompromised.

Medical Uses

If honey had legitimate medical uses, it would arguably be vegan when so used due to medical exceptions.

Commonly, it is used as folk medicine for allergies and cough, and some also believe it has antibiotic properties.

In-vitro, honey does have significant antimicrobial activity due its hypertonic nature (it dries out and kills cells because it draws water away from them). It may have had some limited use in ancient times where more effective topical treatments were unavailable. In-vivo, orally, it has virtually no effect because dilution with water or stomach fluids negate the hypertonic properties.

Topical use in the modern era is more suspect, but certain kinds of honey are occasionally used for wound healing and may have modest evidence. If FDA approved "Medi Honey" which comes from a specific kind of nectar and is specially sterilized is prescribed by a doctor for a burn or post-surgical infection, it is of course vegan.

Beyond that, there's no strong evidence for effect on cough, and there's not even a mechanistic possibility for it to affect common allergies.