High Elves are the classic Lord of the Rings Legolas-fantasy who work with Humans, Dwarves, Gnomes and so on together. It’s a fantasy denied by the developers and has been criticized for decades now. Only the Alliance have them exclusively as a subrace for the Void Elves, but the racials, hairstyles and transmogs does not reflect it well enough.
Blood Elves have forego being High Elves through the events of WC3, officially, and are part of the Horde now. They look like pure High Elves but don’t fit the bill. Only the Alliance does at the moment, because they are the only ones who can use the actual Quel’dorei transmogs, which are Alliance-exclusive.
A recent post also brought the original quote up, that they are no High Elves anymore: