Traditionally, drinking chocolate requires melting chocolate pieces or nibs directly into hot water or milk, while hot chocolate can be crafted by whisking cacao powder with hot milk or water. The latter often allows for a faster preparation, while the former is a ritual, a pause, a moment to cherish.

To be honest there isn't much of a difference besides temperature as chocolate milk can be warmed and called hot chocolate. However, hot chocolate is usually richer and often prepared with real cocoa powder and melted chocolate while chocolate milk is often just regular milk with added chocolate syrup.

Drinking chocolate is different from hot cocoa in that it's made with full-fat chocolate; none of the cocoa butter is removed.