For my case, I grind FATEs so that I could survive dungeons to hunt for items.

Fate:-
Slight faster levelling and GC pts to buy GC items.

Dungeons:-
Fun n item drops.(Crappy due to queuing system)

+++
Exp levels/earnings: its noticed that all my grinding with different class@ same fate same level have different exp gains.

eg. Conj a few hits Vs Thaum spamming : exp gains is slightly higher for Conj.

Other observations:
If Level is 5 levels lesser Fate level, exp gain is minimal or "boss" will reflect a "No Damage" reading.
If Level is 5 levels above Fate level, exp gain is capped@40-50% lesser than previous max exp gained.- skills options are capped too.

Others suggestions,
For dungeons, it shd provide more Gil and GC benefits.

Fate grinding is more for levelling n spamming. =P