New combined finite-differential versions of nonsmooth speed-gradient (SG) algorithms are proposed and examined. Sufficient conditions for stability and robustness of the closed loop system are established. In addition, new passivity definitions suitable for the nonsmooth setting are proposed and passivity of the system with nonsmooth SG algorithms is examined. The proposed finite-differential algorithms possess enhanced dynamic properties and provide extra flexibility for control system design. Particularly, they may operate under a broad uncertainty of plant parameters and disturbances, and they have improved convergence rate and robustness. An illustrative example of adaptive control of Duffing system demonstrates better performance of a combination of smooth and nonsmooth finite-differential SG-algorithms in comparison with performance of the smooth or nonsmooth algorithm.