Think of it like this: The Netherlands is the whole country, like the United States. Holland is just a part of it, like California or Texas.
Holland is actually just two provinces: North Holland (where Amsterdam is) and South Holland (where Rotterdam and The Hague are). These two areas are just a slice of the twelve provinces that make up the Kingdom of the Netherlands.
So, if you call the entire country “Holland,” you’re accidentally ignoring ten other provinces. That’s like calling all of Britain “London” – a bit awkward, right?