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We report a detailed study on the strain-driven phase transition between the tetragonal-like and rhombohedral-like phases in epitaxial BiFeO${}_{3}$ (BFO) thin films which focuses on their structural nature, thermodynamic stability, and ferroelectric/piezoelectric properties. We first show that the tetragonal-like phase, which has a large c/a ratio (\ensuremath{\sim}1.2), in the compressively strained BFO is thermodynamically more favorable at high temperature and high strain state (small thickness). We also report a phase transition between two monoclinic phases at 150 \ifmmode^\circ\else\textdegree\fi{}C. The two monoclinic phases are differentiated by their $c$-axis parameters and tilting angles: The low-temperature phase (M${}_{\mathrm{C}}$) has a $c$-axis parameter of 4.64 \AA{} and a tilting angle (\ensuremath{\beta} $=$ 88.5\ifmmode^\circ\else\textdegree\fi{}) along the $a$ axis, while the high-temperature phase (M${}_{\mathrm{A}}$) has a $c$-axis parameter of 4.66 \AA{} and a tilting angle (\ensuremath{\beta} $=$ 86.8\ifmmode^\circ\else\textdegree\fi{}) along both of the $a$ and $b$ axes. We further show that samples undergoing the M${}_{\mathrm{C}}$--M${}_{\mathrm{A}}$ phase transition exhibit ferroelectric polarization rotation and piezoelectric enhancement. Our findings directly unveil the close links between structural changes, polarization rotation, and large piezoelectricity at morphotropic phase boundaries in BiFeO${}_{3}$.