There is no doubt that French is a beautiful language, probably the most lyrical spoken language in the world. As a native English speaker born in London the sound of someone speaking French has always been a pleasant experience to me. Occasionally a French word and crosses the Channel to make its way into the English; less frequently a French word such as armoire will even establish itself across the other side of the Atlantic too. I suspect that in such cases as the linen armoire much has to do with the fact that it sounds more expensive or impressive than a linen closet or cabinet, and may open the opportunity for charging a premium price.
As far as modern furniture goes an armoire is virtually interchangeable with cabinet, closet or cupboard, depending on what use is to be made of the storage unit and what side of the Atlantic you might be on. A linen cabinet might be called a linen armoire; a TV armoire might be called a computer cupboard. While this all might sound trivial, which it is, it is useful to know that these terms are all used to describe pretty much the same thing. If you want somewhere to store bed linen, towels or table linen then you can do it just as well in a linen cabinet, cupboard, closet or armoire. Why is this worth knowing? If searching for linen storage units online all these search terms are worth searching for as each will throw up a different set of results, although in most cases you will find it difficult to discern the difference between them.
The original French armoire was, as its name suggests, made as a fine piece of furniture to store small arms and ammunition. As that usage faded over the centuries the style of the armoire was not lost, and evolved as a jewelry armoire, linen armoire and more recently a TV or computer armoire. If you really like the look of genuine French antique furniture then with a bit of luck you will find a reproduction for your linen storage. Most likely though it will just be a cabinet, cupboard or closet with a French name that sounds good.