  The branching fractions of the rare weak $B\to \pi l^+l^-(\nu\bar\nu)$ and
$B\to\rho l^+l^-(\nu\bar\nu)$ decays are calculated in the framework of the
relativistic quark model based on the quasipotential approach. The form factors
parametrizing weak decay matrix elements are explicitly determined in the whole
kinematical $q^2$ range without additional assumptions and extrapolations.
Relativistic effects are systematically taken into account including recoil
effects in meson wave functions and contributions of the intermediate
negative-energy states. New experimental data on the differential distributions
in the semileptonic heavy-to-light $B\to\pi l\nu_l$ and $B\to\rho l\nu_l$
decays are analyzed in detail. Good agreement of the predictions and data is
found. The obtained results for the branching fractions of the rare
semileptonic decays are found to be in agreement with other theoretical
estimates and recent experimental data available for the $B^+\to\pi^+
\mu^+\mu^-$ decay.
