The two terms are basically interchangeable, and their use depends largely on local convention.
The equipment they contain includes one or several pumps that raise the fluid so it can continue to flow by gravity. In general, a lift station is characterized by the discharge height, peak flow, and the type of water being conveyed—grey water, wastewater, etc.
Some textbooks differentiate between lift stations and pump stations by whether or not the force main travels horizontally. Both lift stations and pump stations have pumps that move wastewater. Sometimes, wastewater flows into a tank and is pumped out to another gravity sewer line at a higher elevation. In this case, the water is pumped up some distance, but doesn't really travel horizontally very much. This is a LIFT station. Other times, the pumps may move the wastewater over longer distances, sometimes many miles. These are PUMP stations. In truth, we tend to use these terms interchangeably.
There is also something called a booster pump station. These are used in drinking water distribution systems to increase line pressure.
A lift station or a pump station could be a source of pollution if it overflowed. A booster pump station is less likely to be a source of pollution because it is moving clean drinking water. Drinking water contains chlorine, a pollutant that can damage the environment in high enough concentrations, but I don't think that is what is meant in this case.