The differences between the inherent stellar populations (SPs) of LAEs and LBGs are a key factor in understanding early galaxy formation and evolution. We have run a set of SP burst-like models for a sample of 1,558 sources at $3.4<z<6.8$ from the Survey for High-$z$ Absorption Red and Dead Sources (SHARDS) over the GOODS-N field. This work focuses on the differences between the three different observational subfamilies of our sample: LAE-LBGs, no-Ly$alpha$ LBGs and pure LAEs. Single and double SP synthetic spectra were used to model the SEDs, adopting a Bayesian information criterion to analyse under which situations a second SP is required. We find that the sources are well modelled using a single SP in $sim79%$ of the cases. The best models suggest that pure LAEs are typically young low mass galaxies ($tsim26^{+41}_{-25}$ Myr; $M_{mathrm{star}}sim5.6^{+12.0}_{-5.5}times10^{8} M_{odot}$), undergoing one of their first bursts of star formation. On the other hand, no-Ly$alpha$ LBGs require older SPs ($tsim71pm12$ Myr), and they are substantially more massive ($M_{mathrm{star}}sim3.5pm1.1times10^{9} M_{odot}$). LAE-LBGs appear as the subgroup that more frequently needs the addition of a second SP, representing an old and massive galaxy caught in a strong recent star-forming episode. The relative number of sources found from each subfamily at each $z$ supports an evolutionary scenario from pure LAEs and single SP LAE-LBGs to more massive LBGs. Stellar Mass Functions are also derived, finding an increase of $M^{*}$ with cosmic time and a possible steepening of the low mass slope from $zsim6$ to $zsim5$ with no significant change to $zsim4$. Additionally, we have derived the SFR-$M_{mathrm{star}}$ relation, finding a $mathrm{SFR}propto M_{mathrm{star}}^{beta}$ behaviour with negligible evolution from $zsim4$ to $zsim6$.
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