Often times chimneys are not built on a foundation that is connected to the foundation of a home. Gaps can occur between the chimney and exterior wall, or roof overhang are common when the chimney moves relative to the home.
Chimneys will often show signs of settlement sooner than the rest of the home for two reason: First, they are heavy, often two stories high of brick. Second, they often sit on a footing that is not as deep as the rest of the home. So the soil around the chimney footing is affected by environmental factors such as moisture and temperature sooner than the home's footings, which are deper below the surface.
Why should we be worried? Well, the soil that is under the chimney could likely be the same soil that is under our home's footings as well. And although environmental effects on that soil have show up first in the chimney, eventually they could very well impact the deeper soil.
Excerpt from: Foundation Supportworks. Got Tech First ed July 2014, "Chapter 1, Foundation Settlement."