What is residual distortion in a THD analyzer?
The residual distortion is basically the analyzer's internal noise floor - you can't measure anything less than that, because all you'd be measuring is the analyzer's internally-generated noise.  (Your measurement of .0008% says that their specs are fairly conservative, meaning that .004% is likely a worst-case number). 

Think of resolution as being like the number of digits on the display.  In the simplest case, .0001% resolution means that you could (theoretically) discern the difference between a .1000% reading and a .1001% reading.  It has nothing to do with how low you can measure, but only how many digits will be in the answer you get.  The expression "minimum" resolution probably means that this is the best-case resolution, probably available only on one of the lower ranges.  

So, they'll guarantee you can read down to .004%, but with .0001% resolution, so the display should actually read .0040%.  And you'll be able to see the difference between that and .0041%.