Using the File -> Export raw data option you get your hands on the raw data. The file with extension *.WAD contains all the raw data collected in the wave burst mode. The contents are described in the header file, *.HDR. The AST is sampled at twice the rate as the other measurements so you will have to take the data and reconstruct the time series. The values are found in the fields indicated by "AST Distance1 (Beam4)" and "AST Distance2 (Beam4)". This data is likely to have the occasional spikes or outliers. These are removed in the wave processing.
The File -> Export processed data option presents the AST time series after it has been cleaned up with the despiking routines and removing the tidal variation. The despiking is done based on standard deviation and gravity (to read more about this - search for e.g. "Quality"). The AST distance is not compensated for tilt. The format of this is not described, but is quite simply burst counter - sample counter - distance.
So, which one to use? In most cases the despiked data is the best start. If you might want to do the despiking more manually you should start with the raw data.