This paper addresses the challenge of estimating local surface differential properties, specifically surface normals and curvatures, from raw 3D point clouds. Traditional methods either rely on fitting pre-defined analytic surfaces risking model bias, or directly regress normals and curvatures overlooking their intrinsic geometric correlation. We propose a learning-based approach that locally fits osculating implicit quadrics to recover both normals and curvatures simultaneously. Drawing on classical differential geometry, we exploit the fact that every point on a $C^2$ surface admits an osculating quadric in Monge form that exactly reproduces local differential properties. However, the Monge frame itself depends on the very differential quantities being estimated. To bypass this circularity, we reformulate the Monge-form quadric as an implicit representation in a canonical local frame derived solely from point coordinates, enabling supervised learning without requiring Monge frame alignment. This reformulation allows us to construct a ground-truth dataset of such local-frame quadrics and train a neural network to predict per-point weights and offsets for a robust weighted least squares fitting process. The learned offsets account for the deviations of neighboring points from the idealized osculating surface. We further incorporate stable curvature formulations into the training loss alongside normal supervision to enhance estimation fidelity. Extensive experiments on diverse datasets demonstrate that our method outperforms prior approaches in normal and curvature estimation from raw point clouds.