  We study the stellar population properties of 182 spectroscopically-confirmed
(MUSE/VLT) Lyman-$\alpha$ emitters (LAEs) and 450 photometrically-selected
Lyman-Break galaxies (LBGs) at z = 2.8 - 6.7 in the Hubble eXtreme Deep Field
(XDF). Leveraging the combined power of HST and JWST NIRCam and MIRI
observations, we analyse their rest-frame UV-through-near-IR spectral energy
distributions (SEDs) with MIRI playing a crucial role in robustly assessing the
LAE's stellar mass and ages. Our LAEs are low-mass objects
(log$_{10}$(M$_\star$[M$_\odot$]) ~ 7.5), with little or no dust extinction
(E(B - V) ~ 0.1) and a blue UV continuum slope ($\beta$ ~ -2.2). While 75% of
our LAEs are young (< 100 Myr), the remaining 25% have significantly older
stellar populations (> 100 Myr). These old LAEs are statistically more massive,
less extinct and have lower specific star formation rate (sSFR) compared to
young LAEs. Besides, they populate the M$_\star$ - SFR plane along the
main-sequence (MS) of star-forming galaxies, while young LAEs populate the
starburst region. The comparison between the LAEs properties to those of a
stellar-mass matched sample of LBGs shows no statistical difference between
these objects, except for the LBGs redder UV continuum slope and marginally
larger E(B - V) values. Interestingly, 48% of the LBGs have ages < 10 Myr and
are classified as starbursts, but lack detectable Ly$\alpha$ emission. This is
likely due to HI resonant scattering and/or selective dust extinction. Overall,
we find that JWST observations are crucial in determining the properties of
LAEs and shedding light on the properties and similarities between LAEs and
LBGs.
