  In this paper, we experimentally study the influence of large-scale Taylor
rolls on the small-scale statistics and the flow organization in fully
turbulent Taylor-Couette flow {for Reynolds numbers up to $\text{Re}_S=3\times
10^5$}. The velocity field in the gap confined by coaxial and independently
rotating cylinders at a radius ratio of $\eta=0.714$ is measured using planar
{particle image velocimetry} in horizontal planes at different cylinder
heights. Flow regions with and without prominent Taylor vortices are compared.
We show that the local angular momentum transport (expressed in terms of a
Nusselt number) mainly takes place in the regions of the vortex in- and
outflow, where the radial and azimuthal velocity components are highly
correlated. The efficient momentum transfer is reflected in intermittent
bursts, which becomes visible in the exponential tails of the probability
density functions of the local Nusselt number. In addition, by calculating
azimuthal energy co-spectra, small-scale plumes are revealed to be the
underlying structure of these bursts. These flow features are very similar to
the one observed in Rayleigh-B\'{e}nard convection, which emphasizes the
analogies of these both systems. By performing a {complex proper orthogonal
decomposition}, we remarkably detect azimuthally traveling waves superimposed
on the turbulent Taylor vortices, not only in the classical but also in the
ultimate regime. This very large-scale flow pattern{,} which is most pronounced
at the axial location of the vortex center, is similar to the well-known wavy
Taylor vortex flow{,} which has comparable wave speeds, but much larger
azimuthal wave numbers.
