We present the results of a photometric and spectroscopic survey of 321 Lyman break galaxies (LBGs) at z ~ 3 to investigate systematically the relationship between Lya emission and stellar populations. Lya equivalent widths (EW) were calculated from rest-frame UV spectroscopy and optical/near-infrared/Spitzer photometry was used in population synthesis modeling to derive the key properties of age, dust extinction, star formation rate (SFR), and stellar mass. We directly compare the stellar populations of LBGs with and without strong Lya emission, where we designate the former group (EW > 20 AA) as Lya emitters (LAEs) and the latter group (EW < 20 AA) as non-LAEs. This controlled method of comparing objects from the same UV luminosity distribution represents an improvement over previous studies in which the stellar populations of LBGs and narrowband-selected LAEs were contrasted, where the latter were often intrinsically fainter in broadband filters by an order of magnitude simply due to different selection criteria. Using a variety of statistical tests, we find that Lya equivalent width and age, SFR, and dust extinction, respectively, are significantly correlated in the sense that objects with strong Lya emission also tend to be older, lower in star formation rate, and less dusty than objects with weak Lya emission, or the line in absorption. We accordingly conclude that, within the LBG sample, objects with strong Lya emission represent a later stage of galaxy evolution in which supernovae-induced outflows have reduced the dust covering fraction. We also examined the hypothesis that the attenuation of Lya photons is lower than that of the continuum, as proposed by some, but found no evidence to support this picture.