  We present the Differentiable Lensing Lightcone (DLL), a fully differentiable
physical model designed for being used as a forward model in Bayesian inference
algorithms requiring access to derivatives of lensing observables with respect
to cosmological parameters. We extend the public FlowPM N-body code, a
particle-mesh N-body solver, simulating lensing lightcones and implementing the
Born approximation in the Tensorflow framework. Furthermore, DLL is aimed at
achieving high accuracy with low computational costs. As such, it integrates a
novel Hybrid Physical-Neural parameterisation able to compensate for the
small-scale approximations resulting from particle-mesh schemes for
cosmological N-body simulations. We validate our simulations in an LSST setting
against high-resolution $\kappa$TNG simulations by comparing both the lensing
angular power spectrum and multiscale peak counts. We demonstrate an ability to
recover lensing $C_\ell$ up to a 10% accuracy at $\ell=1000$ for sources at
redshift 1, with as few as $\sim 0.6$ particles per Mpc/h. As a first use case,
we use this tool to investigate the relative constraining power of the angular
power spectrum and peak counts statistic in an LSST setting. Such comparisons
are typically very costly as they require a large number of simulations, and do
not scale well with the increasing number of cosmological parameters. As
opposed to forecasts based on finite differences, these statistics can be
analytically differentiated with respect to cosmology, or any systematics
included in the simulations at the same computational cost of the forward
simulation. We find that the peak counts outperform the power spectrum on the
cold dark matter parameter $\Omega_c$, on the amplitude of density fluctuations
$\sigma_8$, and on the amplitude of the intrinsic alignment signal $A_{IA}$.
