Let me preface this by saying I am talking about regular fly rods fished with regular fly lines, not mono rigs or other set-ups...
That being said, even though I own plenty of 3wt rods, the only benefit is that a 3wt line takes up less room on a reel than something heavier. That translates to a smaller reel that balances better with shorter rods which is what all but one of my 3wt rods happens to be.
As far as delicacy goes, unless you are casting beyond 40 feet, there is a negligible difference between a 3wt and heavier lines. In addition when fishing a heavier line you can always fish a longer leader and modify your presentation so line, leader & fly land more gently.
FWIW - I never fish anything lighter than a 6wt or 5wt line on the Letort & Big Springs...
A lot of anglers today are terrified of anything heavier than a 4wt because they think trout have evolved way beyond the point when guys caught them with regularity on 6 & 7wt outfits...
Let me also add that there are choices in 4wt - 7wt rods that are not super fast broomsticks if you think that only the lightweight stuff is suitable for feeling the tug of smaller fish.
Finally ask yourself this question too which is VERY important if you think lighter means more delicate:
Does the 3wt rod a person is fishing load when using a 3wt fly line that weighs 100 grains in the first 30 yards (a true-to-weight 3wt line) or does the rod need a 1/2 or whole size heavier 3wt line to feel the way they want?
If the answer is that the rod needs a heavier 3wt line...
You are fishing a 3wt line in name only.
Good luck!