  We present a comparative study of inflation in two theories of quadratic
gravity with {\it gauged} scale symmetry: 1) the original Weyl quadratic
gravity and 2) the theory defined by a similar action but in the Palatini
approach obtained by replacing the Weyl connection by its Palatini counterpart.
These theories have different vectorial non-metricity induced by the gauge
field ($w_\mu$) of this symmetry. Both theories have a novel spontaneous
breaking of gauged scale symmetry, in the absence of matter, where the
necessary scalar field is not added ad-hoc to this purpose but is of geometric
origin and part of the quadratic action. The Einstein-Proca action (of
$w_\mu$), Planck scale and metricity emerge in the broken phase after $w_\mu$
acquires mass (Stueckelberg mechanism), then decouples. In the presence of
matter ($\phi_1$), non-minimally coupled, the scalar potential is similar in
both theories up to couplings and field rescaling. For small field values the
potential is Higgs-like while for large fields inflation is possible. Due to
their $R^2$ term, both theories have a small tensor-to-scalar ratio ($r\sim
10^{-3}$), larger in Palatini case. For a fixed spectral index $n_s$, reducing
the non-minimal coupling ($\xi_1$) increases $r$ which in Weyl theory is
bounded from above by that of Starobinsky inflation. For a small enough
$\xi_1\leq 10^{-3}$, unlike the Palatini version, Weyl theory gives a
dependence $r(n_s)$ similar to that in Starobinsky inflation, while also
protecting $r$ against higher dimensional operators corrections.
