  (Abridged) We extend our previous studies aimed at modeling Lyman Alpha
Emitters (LAEs) to the second population of high redshift sources, Lyman Break
Galaxies (LBGs), with the final goal of investigating the physical relationship
between them at 6 < z <8. We use a set of large (~10^6 Mpc^3) cosmological SPH
simulations that include a detailed treatment of star formation, feedback,
metal enrichment and supernova dust production; these same simulations have
already been shown to successfully reproduce a large number of observed
properties of LAEs (Dayal et al. 2010). We identify LBGs as galaxies with an
absolute ultraviolet (UV) magnitude M_{UV} <= -17, consistent with current
observational criteria. With no further parameter tuning, the model reproduces
the redshift evolution of the LBG UV luminosity function, stellar mass
function, and star formation rate (SFR) density extremely well, compared to the
data at z~6-8. It predicts a z-independent average beta value of about -2.2,
consistent with the most recent observational results. The mean LBG specific
SFR increases from 6.7 Gyr^-1 at z ~ 6 to 13.9 Gyr^-1 at z ~ 8, and is largest
for the smallest LBGs. From a comparison of the simulated LAE and LBG
populations, we find no appreciable differences between them in terms of the
stellar masses, ages, SFR, and dust content; only the faintest LBGs with M_{UV}
>= -18 (-19) at z ~ 6 (8) do not show an observable Lyman Alpha line. LAEs
hence represent a luminous LBG subset, whose relative extent depends only on
the adopted equivalent width (EW) selection criteria. Further, in line with
current observations, we find that the fraction of LBGs showing a Lyman Alpha
line decreases with increasing redshift due to the combined effects of dust and
reionization. We conclude with a brief critical model discussion, which
emphasizes the uncertainties inherent to theoretical EW determinations.
