Not an expert, but here is my answer for your question:
Vilem = "would like" is a more calm, polite, and social form of wanting. "I would kindly like a glass of water"
Cupio = "want" is a more passionate desire. "I'm so thirsty after that journey, I want a glass of water or I might pass out." It is a more covetus, intense want.
Perhaps the Romans used their word for "want" only when it was a very important want, and they used their word for "would like" to indicate a simple or trivial want.