I like Neuschwanstein. It's sort of an icon. My ex liked Hohenschwangau better because it was an actual home.
Neither is technically a castle (Burg); they are palaces (Schlößer). Hohenschwangau was built in around 1800 over the ruins of a real castle by Ludwig's father, king of Bavaria, as a fortified home for his family. Neuschwanstein was built by Ludwig as his palace, having the outward appearance of a real castle, but just for his living quarters. It was built around the same time as the Eifel Tower and the Brooklyn Bridge.
But the distinction between castles and palaces is quite vague. There are plenty of palaces, private homes, hostel and hotels (Reichsburg in Cochem, Stahleck in Bacharach, Hohenzollernburg in Hechingen) that are built over old castle ruins to look like castles, but are not. I would question whether Burg Eltz is properly a castle and not a residence. To see an actual castle, go to the Marksburg, Harburg, or Burghausen, to name a few.
So understand the difference and enjoy either one. Hohenschwangau was Ludwig's boyhood home; Neuschwanstein was his fantasy.