LWC and Absolute Humidity
According to the online chapter Water Vapor (University of British Columbia), the two variables you ask about are defined as:

Absolute humidity

$$\rho_{v}=\frac{m_{water\ vapour}}{volume}$$

Liquid water content

$$\rho_{LWC}=\frac{m_{liquid\ water}}{volume}$$

The main difference is that absolute humidity measures the mass of water vapour and the LWC measures the mass of suspended or falling liquid water content in a unit volume of air. A handy definition linking the two terms is (from the linked document):


  LWC is the liquid-water analogy to the absolute
  humidity for water vapor.