What is the difference between (responsibilities wise) police and sheriff department?

Geography, mostly. They serve different jurisdictions. Police are usually city or metro, sheriff is usually county or parish. Police usually have a small jail, sheriff usually runs the county jail.

Warrants, courtroom duty and that sort of thing usually go to the sheriff.

- Stuart

A lot depends on what state you are in. In many northeastern states, Sheriffs and their deputies have limited law enforcement powers. In other states, they have all the same authority as any law enforcement officer. Sheriffs are "county" officers, and their jurisdiction includes the entire county. Police are USUALLY municipal law enforcement officers, and their jurisdiction is limited to their own city. However, their are some place that have 'regional' police departments, the encompass several municipalities. And there are Counties that have "County Police", which are separate from the County Sheriff. There is no single answer to your question, unless you specify one particular state.

Police has jurisdiction in city limits, and the sheriff is in the county.

NOTHING. Police are hired by the CITY/TOWN/VILLAGE to enforce the laws. The Sheriff is hired by the COUNTY to enforce the law. The RESPONSIBILITIES are the same. Only the JURISDICTION is different.