Two initial observations : 

1. Are the darts launched with the same speed or the same kinetic energy?

When you specify that you use the same 'force' to launch both darts, it is not clear what you intend. Using the same force in Newtons over the same distance should result in the two darts acquiring the same kinetic energy but different speeds. However, as BowlofRed points out, your ability to launch objects is limited by the mass of your arm. If the mass of the arm is much larger than the mass of the dart, very little KE is imparted to the dart. All light-weight objects are launched with approximately the same speed, which is limited by size and strength of the arm and the technique used.  

2. The effect of gravity is a complicating factor here, so let us ignore it.

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For the same shape and size of object, travelling at the same speed, the drag force $F$ caused by air resistance is the same. It depends on cross-sectional area but not mass. For thrown objects, drag force is approximately proportional to speed through the air : $F=-kv$.

**If the darts are launched with the same KE then they travel the same distance by the time they have lost all their KE.** This is because the same force retards them both, and it does the same work to stop them, so the must act over the same distance.
 
**If the darts are launched with the same speed then the heavier dart travels further.** This is easily seen by comparison with the case of the same KE. The lighter dart has less KE, the same force retards both, so its KE is expended over a shorter distance than for the heavier dart.