  We report a detailed analysis of the energy spectra, second- and high-order
structure functions of velocity differences in superfluid $^4$He counterflow
turbulence, measured in a wide range of temperatures and heat fluxes. We show
that the one-dimensional energy spectrum $E_{xz} (k_y)$ (averaged over the
$xz$-plane, parallel to the channel wall), directly measured as a function of
the wall-normal wave-vector $k_y$, gives more detailed information on the
energy distribution over scales than the corresponding second-order structure
function $S_{2}(\delta_y)$. In particular, we discover two intervals of $k_y$
with different apparent exponents: $E_{xz} (k_y)\propto k_y^{-m_C}$ for
$k\lesssim k_\times$ and $E_{xz} (k_y)\propto k_y^{-m_F}$ for $k\gtrsim
k_\times$. Here $k_\times$ denotes wavenumber that separate scales with
relatively strong (for $k\lesssim k_\times$) and relatively weak (for $k\gtrsim
k_\times$) coupling between the normal-fluid and superfluid velocity
components. We interpret these $k$-ranges as cascade-dominated and mutual
friction-dominated intervals, respectively. General behavior of the
experimental spectra $E_{xz}(k_y)$ agree well with the predicted spectra [Phys.
Rev. B 97, 214513 (2018)]. Analysis of the $n$-th order structure functions
statistics shows that in the energy-containing interval the statistics of
counterflow turbulence is close to Gaussian, similar to the classical
hydrodynamic turbulence. In the cascade- and mutual friction-dominated
intervals we found some modest enhancement of intermittency with respect of its
level in classical turbulence. However, at small scales, the intermittency
becomes much stronger than in the classical turbulence.
