In American English, we usually only say "cupboard" when we're talking about a place where food is kept, but you will also hear people say "kitchen cabinets." Use whichever word you prefer. If your listener isn't sure what you mean, you can give them more detail to clarify your intended meaning.

I would say they tend to blur together. But a cupboard is generally used to store kitchen objects, be it dishes or food, or other things one might put in a kitchen. Cabinets can be found in the kitchen or anywhere else, and can hold anything. Maybe cupboard is slightly less common in American English, but it is used a lot just the same. A kitchen cabinet is built in, though such objects could also be called a cupboard. If you have a free-standing piece of kitchen furniture, that should not be called a cabinet. Cabinets can also come in other varieties such as file cabinets.

I'm a US speaker. They are close synonyms. To me, a cupboard is a _kind_ of cabinet. A cupboard is usually found in a kitchen or a dining room, and usually contains--as the word suggests--cups, or plates, spoons, knives... or food. Cupboards can be called cabinets. In our dining room we have a piece of wooden freestanding furniture with glass doors, shelves, and drawers, and we call it a "cabinet." We could also call it a "cupboard," but we don't. In contrast, in my office there is a piece of metal furniture with file folders hanging in it, and it is a "file cabinet." Nobody would call it a "file cupboard." You probably know that "cupboard" is pronounced "cubbard." A famous nursery rhyme tells us Old Mother Hubbard Went to the cupboard To get her poor dog a bone; But when she got there The cupboard was bare And so the poor dog had none. This rhyme illustrates both the pronunciation of "cupboard" and its use to store food. (However, in present-day English, "none" does not rhyme with "bone.")

Cabinets are a type of built-in storage, usually, often in a kitchen or a bathroom. To install cabinets, you hire a cabinet maker who is usually a carpenter who makes the cabinets out of wood. You never install just one cabinet. You install "cabinets". They can have doors on hinges and sliding drawers. What makes cabinets "cabinets" is not the purpose for which they are used. It is a type of built-in furniture that you can use for any purpose whatsoever. There is no such thing as a "cupboard maker". Unlike cabinets, cupboards need not be built-in, though they might have built-in components. A cupboard can be a free standing piece of furniture. A cupboard can even be an entire room and that room can even contain cabinets. "Cupboard" does not describe any particular physical object. It is a storage area for food and random kitchen objects like brooms and dust pans normally adjacent to the kitchen. The name of such a room can be either the "cupboard" or the "kitchen closet", but the former is preferred if it is a food storage area. If the room holds only brooms, buckets, and trash cans then you would call it the "kitchen closet". It is definitely cool to have a cupboard. Having a cupboard is characteristic of old houses.