While extended research has been conducted on solid particles settling in turbulence, most of them have focused on very dense particles like solid ones in air turbulence, or on almost neutrally buoyant ones. The intermediate case of particles whose density is representative of solid particles falling in water turbulence has been scarcely explored in the academic literature. In the present work, we aim to bridge this gap. To this end, we perform Eulerian-Lagrangian point-particle numerical simulations to examine the settling of small particles in homogeneous turbulence with two values of particle-to-fluid density ratio, $\Gamma =4$ and $14$. The analysis of the results provides a comparison with the much more investigated case of very heavy particles with $\Gamma =1000$ to compare and contrast the settling mechanisms between the moderate and large density ratios. The simulations are performed in one-way and two-way coupling for dilute particle-laden conditions and two turbulence Reynolds numbers, $R_{\lambda }=40$ and $115$. In two-way coupling, both particle self-disturbance and force-spreading operator related issues are handled by determining an effective particle diameter. The main results show that the settling increase is significantly smaller for moderate than large values of $\Gamma$ and that the commonly reported centrifuge effect for high density ratios is not observed for the much smaller ones considered here. Under two-way coupling, it is found that for the moderate density ratios, the particle settling velocity is not mainly governed by the sampled fluid velocity as a consequence of a significant added mass contribution that tends to balance the dragged fluid motion. The two considered values of the turbulence Reynolds number do not have a significant effect on the analyzed particle and flow quantities. Finally, as far as the existing literature allows for comparison, the experimental data used for comparisons of the settling velocity, fluid and particle velocity and acceleration statistics are in fair agreement with the simulations.