We measure the evolution of the rest-frame UV luminosity function (LF) and the stellar mass function (SMF) of Lyman-α (Ly α) emitters (LAEs) from z ~ 2 to z ~ 6 by exploring ~4000 LAEs from the SC4K sample. We find a correlation between Ly α luminosity (LLy α) and rest-frame UV (MUV), with best fit M$_{\rm UV}=-1.6_{-0.3}^{+0.2}\log _{10} (\rm L_{Ly\,\alpha }/erg\, s^{-1})+47_{-11}^{+12}$ and a shallower relation between LLy α and stellar mass (M⋆), with best fit $\log _{10} (\rm M_\star /{\rm M}_\odot)=0.9_{-0.1}^{+0.1}\log _{10} (\rm L_{Ly\,\alpha }/erg\, s^{-1})-28_{-3.8}^{+4.0}$. An increasing LLy α cut predominantly lowers the number density of faint MUV and low M⋆ LAEs. We estimate a proxy for the full UV LFs and SMFs of LAEs with simple assumptions of the faint end slope. For the UV LF, we find a brightening of the characteristic UV luminosity (M$_{\rm UV}^*$) with increasing redshift and a decrease of the characteristic number density (Φ*). For the SMF, we measure a characteristic stellar mass (${\rm M_\star ^*/{\rm M}_\odot }$) increase with increasing redshift, and a Φ* decline. However, if we apply a uniform luminosity cut of $\log _{10} (\rm L_{Ly\,\alpha }/erg\, s^{-1}) \ge 43.0$, we find much milder to no evolution in the UV and SMF of LAEs. The UV luminosity density (ρUV) of the full sample of LAEs shows moderate evolution and the stellar mass density (ρM) decreases, with both being always lower than the total ρUV and ρM of more typical galaxies but slowly approaching them with increasing redshift. Overall, our results indicate that both ρUV and ρM of LAEs slowly approach the measurements of continuum-selected galaxies at z > 6, which suggests a key role of LAEs in the epoch of reionization.