Kerosene and heating oil -- more formally called No. 2 fuel oil -- are both combustible hydrocarbon liquids, but they are chemically different because they result from different steps in the refining of crude oil. As a result, the two are generally used for different purposes.
As chemical substances, kerosene and No. 2 fuel oil have different maximum distillation temperatures: 204 degrees Celsius for kerosene and 338 degrees Celsius for No. 2 fuel oil. They also have different "flash points," or temperatures at which it's possible for the substances to ignite when mixed with air: the lowest is 37.8 degrees Celsius for kerosene and 52.2 degrees Celsius for No. 2 fuel oil.
ASTM International, which publishes industrial standards for worldwide use, classifies kerosene and fuel oil No. 2 as distinct materials. Kerosene must meet the ASTM D3699 standard, while No. 2 fuel oil is defined by ASTM D396.
For more than a century, kerosene has been used as a light source when burnt in small lamps. It is also used domestically in cook stoves and space heaters. No. 2 fuel oil is also used in domestic settings, but typically burnt to heat interior spaces, hence the common name "heating oil." Sometimes No. 2 fuel oil is burnt to heat smaller commercial spaces as well.