The definition of a sheriff's department can vary, for example, here in South Carolina, a sheriff is the director of a county-wide law enforcement agency. A municipal agency cannot be a "sheriff's office" or a "sheriff's department". I'm not familiar with how it works anywhere else, but here, the sheriff's deputies have full authority throughout the county, but are simply less present in municipal areas that have their own law enforcement agency. But, that is specifically South Carolina. I'm not familiar with public safety law anywhere else, so I can't provide examples from anywhere else.