It's complicated but simply put - there are multiple verbs for 'wearing' in Vietnamese depending on the object: mặc, mang, thắt, đeo and đội.
mặc: this is the generic verb and used with clothing itself (mặc quần áo = to wear clothes/clothing). You use it for anything covering your torso or legs like shirts, coats, pants a.s.o.
mang: generally means to carry but here it's used for shoes and socks. (e.g. mang giày = to wear shoes, mang tất/vớ = to wear socks)
thắt: things you fasten like a tie, scarf or belt (e.g. thắt ca vát/cà vạt = to wear a tie, thắt dây lưng/dây nịt = to wear a belt, thắt khăn choàng cổ = to wear a scarf)
đeo: for jewellery and other accessories like glasses (e.g. đéo kính/kiếng = to wear glasses, đeo bông tai = to wear earrings, đeo nhẫn = to wear a ring)
đội: things you place onto your head like hats. (e.g. đội mũ/nón = to wear a hat)
In my experience, "mang" can go with any type of clothing, but it's kind of rare in northern dialect. In northern dialect (more precisely, Hanoi dialect), people use:
- mũ (hats): đội
- khăn (scarves): quàng, choàng. Exception: đeo khăn quàng đỏ
- áo (top, shirt), quần (pants): mặc
- găng tay (gloves): đi, đeo
- giày dép (shoes), tất (socks): đi
- mặt nạ (mask), khẩu trang (surgical mask): đeo
- nhẫn (ring): đeo
I'm not sure about the south. Probably they use "mang" more often.
It's not wrong to use it, so if you are not sure which verb to use, just use "mang".
Just as a quick random lesson, Vietnamese actually utilises 5 different words to mean "wear" depending on the item of clothing or accessory.
mặc: the generic verb for things you wear on your torso and legs such as shirts and pants. E.g. mặc quần (to wear pants), mặc áo (to wear a shirt).
mang: use this when referring to things you slip on such as gloves and socks. E.g. mang găng/bao tay (to wear gloves), mang tất/vớ (to wear socks)
đội: use this for headwear. E.g. đội mũ/nón (to wear a hat)
đeo: use this for accessories; jewellery. E.g. đeo kính/kiếng (to wear glasses), đeo hoa/bông tai (to wear earrings)
thắt: use this for things you fasten onto yourself. E.g. thắt dây lưng/nịt (to wear a belt), thắt quàng cổ/choàng cổ (to wear a scarf)
Now obviously there are exceptions and you can also say mặc áo khoác and mang can also mean to bring (although generally through the combination of mang theo).