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 this is larger than the masses of the darts, the combined kinetic energy of arm plus dart may be approximately the same in both cases, so that the darts are launched with approximately the same speed.

**1. Launch with same Kinetic Energy**

If, instead of throwing, you use the rail gun or catapult or something similar to launch the darts, then both the heavy and light darts will be launched with the same kinetic energy : $\frac12mV^2=\frac12Mv^2$. The lighter dart will have the higher launch velocity. 

If there is no air resistance, the dart with the higher launch velocity will travel further. How far a projectile goes does not depend on its mass, only launch speed and angle and the strength of gravity. For a given launch angle, the [range of a projectile](https://en.wikipedia.org/wiki/Range_of_a_projectile) is proportional to $v^2$, so the ranges of the 2 darts would be in inverse ratio of their masses :  
$R_m/R_M=V^2/v^2=M/m$.

If there is air resistance it is proportional to speed, unless the speed through the air is high. It also depends on cross-sectional area, but these are assumed to be the same. It does not depend on mass. So the drag force on the lighter dart is higher. However, the deceleration of the lighter dart is higher, because force is higher and mass is smaller : $a=F/m$. So the ratio of decelerations would be $a_m/a_M=(V/v)(M/m)$. The deceleration of the lighter mass would be much higher.  

If the launch angle is small, the velocity is mostly horizontal. The range is approximately $v^2/2a$. So the ratio of ranges is  
$R_m/R_M=(V/v)^2(a_M/a_m)=(V/v)^2(v/V)(m/M)=mV/Mv=\sqrt{M/m}.$  
The **lighter dart should travel further**, although not in direct proportion to the masses as in the case without air resistance. 

**2. Launch with same Speed**

Now in the absence of air resistance the ranges would be the same. Putting $V=v$ in the final result above, the ratio of ranges with significant air resistance would be :  
$R_m/R_M=mV/Mv=m/M$.  
In this case, **the heavier dart travels further**.