  For the simulation of equilibrium states and finite-temperature response
functions of strongly-correlated quantum many-body systems, we compare the
efficiencies of two different approaches in the framework of the density matrix
renormalization group (DMRG). The first is based on matrix product
purifications. The second, more recent one, is based on so-called minimally
entangled typical thermal states (METTS). For the latter, we highlight the
interplay of statistical and DMRG truncation errors, discuss the use of
self-averaging effects, and describe schemes for the computation of response
functions. For critical as well as gapped phases of the spin-1/2 XXZ chain and
the one-dimensional Bose-Hubbard model, we assess the computation costs and
accuracies of the two methods at different temperatures. For almost all
considered cases, we find that, for the same computation cost, purifications
yield more accurate results than METTS -- often by orders of magnitude. The
METTS algorithm becomes more efficient only for temperatures well below the
system's energy gap. The exponential growth of the computation cost in the
evaluation of response functions limits the attainable timescales in both
methods and we find that in this regard, METTS do not outperform purifications.
