We provide a comparative study between two different approaches to construct nonlinear filter estimators: on the one hand grid methods using zero-order and first-order quantization schemes, and on the other hand particle filtering algorithms using sequential importance sampling or resampling. For each method, numerical implementation is explicited in addition to convergence arguments and algorithmic complexity. Numerical examples are then given over three state space models: the Kalman filter case, the canonical stochastic volatility model and the infinite dimension explicit filter introduced in [Genon-Catalot, V., 2003, A non linear explicit filter. | 
