A wardrobe is usually a tall cupboard that has clothes hanging in it, and possibly has shelves as well to put clothes or accessories in, shoes can also go there. This can be freestanding or accessed via a door/ a cupboard. It is usually in a bedroom, but not always. A cupboard is usually accessed through a door and it can be like any other door in your home, or a cupboard can actually be various shapes and sizes. It can contain clothes etc., or it can contain no clothes. It can contain books, belongings or anything; cupboards can be in a kitchen, bedroom or bathroom etc. Drawers can be in a ‘chest of drawers’ (this latter phrase usually only contains clothing items), which can often contain clothes and accessories, though drawers can also be in any room in the house/garage. It often contains smaller items than in cupboards such as socks or cutlery.

A wardrobe is a cupboard for clothes. A cupboard opens via a door. Drawers open via sliding.

they are synonyms

Just something to be aware of : these are British definitions of British terms. Americans use other terms - such as 'closet' - and an American would therefore give you different interpretations of these words.