I would say "The Nederlands" (In Dutch: Nederland) is always used in formal situations. In informal situations, "The Netherlands/Nederland" wins too but people care less. It depends a bit on where people are from. Frysians and Limburgers hardly ever will use "Holland" except at soccer matches, because it shouts better. I as a Hollander myself probably only use "Holland" when speaking English, especially to those that have local names based on Holland, like Italians, where they use Olanda. I think the rule is: in English we hardly care, in Dutch, quite some people do, but most people say "Nederland" anyway.
Well, it's clear that "Holland" and its equivalents in other languages have become widespread (another example is Japanese: oranda), but I don't know why that would suggest there is more to why Dutch people consider it wrong or at least weird than the explanation given by Laurens above. It just shows that this pars-pro-toto word for it has, yes, become widespread.
In Spanish we have two different terms, "Holanda" and "Países bajos", when we talk about "Los Países bajos" we generally understand you're talking about the Benelux, whereas Holanda is just used for the Netherlands, anyway, both terms are also used just for the Netherlands too, but "Holanda" is kind of more specific
That's weird, because Luxemburg and a large part of Belgium is quite mountainous. Also, in French Les Pays Bas, in German Die Niederlande, in English The Netherlands, in Italian Paesi Bassi all literally mean the same thing: The Netherlands (The low lands, Nether being an old form of Low). Not the Benelux. So why suddenly in Portuguese it means something else?
I've occasionally, though admittedly rarely, in my travels heard the US called by one of the states; California being the one I most often recall. Since I was from Alaska for most of that time (currently residing in Colorado FYI) I wasn't too enamored of the designation, but I could understand it. It made sense to somebody.
Something more about that: Even though the language is called Dutch (in English) or Nederlands (in Dutch), in Bulgaria it's commonly called something like Hollandish (холандски). I've researched that as I had the same question as you do, and it seems that just as @LaurensEduard, Holland is a province (actually two) of the Nederlands, and the country actually has several ethnic groups. Calling the language Hollandish will be offending to most of them who don't really live in Holland and are not of the same ethnicity. Same goes for the country name.
I figured that if I had a question, then I wasn't going to be the only one to have it. Thank you for the input.
Just because the Y language name for a country is X doesn't mean that X-ish is considered complementary to the folks who live there. We know that England and the Netherlands have a long, mutual, history of economic and military dispute with each other, most of which predates the American independence.
People who live in, or near, the two provinces which are in the west and called North and South Holland, often refer to their country as Holland, and residents as Hollanders, who speak Hollands. However, people who live in other parts of the country, tend to dislike being associated with the term Holland and prefer The Netherlands, Nederlanders and Nederlands as language. Especially in the most southern province, Limburg, there's a competative dislike against anything "Hollands" because people from the west of the country often make fun of the accent of fellow Dutchmen in Limburg (which is strongly influenced by German and Flemish). So, although Holland is widely accepted, the most political correct name would definately be The Netherlands.