  We present a study of the intersection between the populations of star
forming galaxies (SFGs) selected as either Lyman break galaxies (LBGs) or
Lyman-alpha emitters (LAEs) in the redshift range 2.9 - 6.7, within the same
volume of universe sampled by the Multi-Unit Spectroscopic Explorer (MUSE)
behind the Hubble Frontier Fields lensing cluster A2744. We define three
samples of star-forming galaxies: LBG galaxies with an LAE counterpart (92
galaxies), LBG galaxies without LAE counterpart (408 galaxies) and LAE galaxies
without an LBG counterpart (46 galaxies). All these galaxies are intrinsically
faint due to the lensing nature of the sample (Muv $\ge$ -20.5). The fraction
of LAEs among all selected SFGs increases with redshift up to z $\sim$ 6 and
decreases for higher redshifts. The evolution of LAE/LBG populations with UV
magnitude and Lya luminosity shows that the LAE selection is able to identify
intrinsically UV faint galaxies with Muv $\ge$ -15 that are typically missed in
the deepest lensing photometric surveys. The LBG population seems to fairly
represent the total population of SFGs down to Muv$\sim$-15. Galaxies with
Muv$<-17$ tend to have SFRLya$<$SFRuv, whereas the opposite trend is observed
within our sample for faint galaxies with Muv$>-17$, including galaxies only
detected by their Lya emission, with a large scatter. These trends, previously
observed in other samples of SFGs at high-$z$, are seen here for very faint
Muv$\sim -15$ galaxies, much fainter than in previous studies. There is no
clear evidence, based on the present results, for an intrinsic difference on
the properties of the two populations selected as LBG and/or LAE. The observed
trends could be explained by a combination of several facts, like the existence
of different star-formation regimes, the dust content, the relative
distribution and morphology of dust and stars, or the stellar populations
