Lamp oil and kerosene are in the same family of oils but the difference between lamp oil and kerosene is that paraffin lamp oil has been refined much more to allow the lamp oil to burn more cleanly than kerosene.
Kerosene isn't as highly refined and burns less clean than lamp oil does.
Lamp oil is also a bit more expensive than straight kerosene due to the much more involved process of refining and purifying the lamp oil.
Kerosene doesn't have to be purified like the lamp oil does.