Z and Up are both measurements of vertical velocity, but with different reference points.

• Z always points directly out from the centre of the ADCP head, and will measure different planes depending on how the ADCP is orientated. For example, if the ADCP is tilted 10°, Z will measure a plane 10° from vertical.
• Up is referenced to the Earth, and points upwards regardless of how the ADCP is orientated in space. For example, if the ADCP is tilted 10°, Up will still measure vertical velocity.

Figure: The Z axis is in reference to the instrument head and changes with tilt. The Up-axis is relative to the Earth and is not affected by tilt. 

You can find Z and Up by using the Right-Hand Rule, with the positive directions given by your thumb. More details about coordinate system can be found here: What are the different coordinate systems and how are they defined?