The finite-temperature behavior of small silicon and tin clusters (${\mathrm{Si}}_{10}$, ${\mathrm{Si}}_{15}$, ${\mathrm{Si}}_{20}$, ${\mathrm{Sn}}_{10}$, and ${\mathrm{Sn}}_{20}$) is studied using isokinetic Born-Oppenheimer molecular dynamics. We find that the low-lying structures for all the clusters are built upon a highly stable tricapped trigonal prism unit which is seen to play a crucial role in the finite-temperature behavior. The thermodynamics of small tin clusters is revisited in light of the recent experiments on tin clusters of sizes 18--21 [G. A. Breaux et al., Phys. Rev. B, 71, 073410 (2005)]. Our calculated heat capacities for ${\mathrm{Si}}_{10}$, ${\mathrm{Sn}}_{10}$, and ${\mathrm{Si}}_{15}$ show main peaks around 2300, 2200, and $1400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, respectively. The finite-temperature behavior of ${\mathrm{Si}}_{10}$ and ${\mathrm{Sn}}_{10}$ is dominated by isomerization and it is rather difficult to discern their melting temperatures. On the other hand, ${\mathrm{Si}}_{15}$ does show a liquidlike behavior over a short temperature range, which is followed by fragmentation observed around $1800\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The finite-temperature behavior of ${\mathrm{Si}}_{20}$ and ${\mathrm{Sn}}_{20}$ shows that these clusters do not melt but fragment around 1200 and $650\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, respectively.