  In this study, our primary objective is to compare the properties of SMBH and
their host galaxies between type 1 and type 2 AGN. In our analysis, we use
X-ray detected sources in two fields, namely the eFEDS and the COSMOS-Legacy.
To classify the X-ray sources, we perform SED fitting analysis, using the
CIGALE code. Ensuring the robustness of our analysis is paramount, and to
achieve this, we impose stringent selection criteria. Thus, only sources with
extensive photometric data across the optical, near- and mid-infrared part of
the spectrum and reliable host galaxy properties and classifications were
included. The final sample consists of 3,312 AGN, of which 3\,049 are
classified as type 1 and 263 as type 2. The sources span a redshift range of
$\rm 0.5<z<3.5$ and encompass a wide range of L$_X$, falling within $\rm
42<log,[L_{X,2-10keV}(ergs^{-1})]<46$. Our results show that type 2 AGN exhibit
a tendency to inhabit more massive galaxies, by $0.2-0.3$\,dex, compared to
type 1 sources. Type 2 AGN also display, on average, lower specific black hole
accretion rates, a proxy of the Eddington ratio, compared to type 1 AGN. These
differences persist across all redshifts and L$_X$ considered within our
dataset. Moreover, our analysis uncovers, that type 2 sources tend to have
lower star-formation rates compared to 1 AGN, at $\rm z<1$. This picture
reverses at $\rm z>2$ and $\rm log,[L_{X,2-10keV}(ergs^{-1})]>44$. Similar
patterns emerge when we categorize AGN based on their X-ray obscuration levels
($N_H$). However, in this case, the observed differences are pronounced only
for low-to-intermediate L$_X$ AGN and are also sensitive to the $\rm N_H$
threshold applied for the AGN classification. These comprehensive findings
enhance our understanding of the intricate relationships governing AGN types
and their host galaxy properties across diverse cosmic epochs and luminosity
regimes.
