  Recently, DESI has released baryon acoustic oscillation (BAO) data, and DES
has also published its five-year supernova (SN) data. These observations,
combined with cosmic microwave background (CMB) data, support a dynamically
evolving dark energy at a high confidence level. When using cosmological
observations to weigh neutrinos, the results of weighing neutrinos will be
significantly affected by the measurement of dark energy due to the degeneracy
between neutrino mass and the dark-energy equation of state. Therefore, we need
to understand how the dynamical evolution of dark energy in the current
situation will affect the measurement of neutrino mass. In this work, we
utilize these latest observations and other additional distance measurements to
discuss the mutual influence between neutrinos and dark energy, then calculate
the Bayes factor to compare models. We consider three neutrino mass hierarchies
including degenerate hierarchy (DH), normal hierarchy (NH), and inverted
hierarchy (IH), as well as three dark energy models including $\Lambda \rm
CDM$, $w\rm CDM$, and $w_0w_a \rm CDM$ models. Cosmological data combined with
the prior of particle physics experiments can provide strong to decisive
evidence favoring the $w_0w_a {\rm CDM}+\sum m_\nu$ model with NH. In the
$w_0w_a \rm CDM$ model, using the CMB+DESI+DESY5 data, we obtain constraints on
the total neutrino mass, $\sum m_\nu<0.171\ \rm eV,\ 0.204\ \rm eV,\ 0.220\ \rm
eV$, for DH, NH, and IH, respectively. Furthermore, taking into account the
neutrino hierarchy or incorporating additional distance measurements results in
a more pronounced deviation from the $\Lambda$CDM model for dark energy. The
latter, particularly, exhibits a deviation at a confidence level that surpasses
$4\sigma$.
