Monschau is the Eifel at its most postcard-like: slate roofs, half-timbered houses, cobbled lanes, and the Rur squeezed between steep green slopes. It is small enough to cover on foot, but it has the depth that many pretty towns lack. The old cloth-trade wealth left behind serious houses, the castle keeps watch from the hill, and the surrounding country is made for short hikes and bike rides.
Tourism has not discovered Monschau quietly. Summer weekends and the Christmas market can be crowded, and parking is often the least romantic part of the visit. Come midweek or outside peak hours and the town makes much more sense: a compact old centre, a couple of strong museums and craft stops, and easy access to Eifel trails.
Monschau works best as a day from Aachen or as a slower overnight stop if you are crossing the German-Belgian borderlands. It is not a city of big sights, but the whole setting is strong enough to justify the detour.