What is the difference between a grand auditorium and a grand symphony guitar?

Well your auditorium is more bright sounding great for fingerpicking, they also tend to be smaller.

The symphony guitar is rich lively sound similar but not as bell like as the auditorium.

while im nameing these body styles.....

Dreadnought guitar- mostly used in country/bluegrass music there typicaly have bigger bodies and have deep rich warm tones.

Your Concert Guitars are usually loud. well the name says it all there also bigger guitars to produce more tone.