The difference is not really about the number of variables but the dynamics of the plant... The classical formulation of Kalman filters assumes gaussian distri

The difference is not really about the number of variables but the dynamics of the plant... The classical formulation of Kalman filters assumes gaussian distributions for the variables. This makes the computations much faster since the posterior distributions is a few matrix operations away. This works fine when the model for the plant is linear (since a linear transformation of a gaussian variable is gaussian). In situations where the plant model is not linear estimates can go awry quickly. In the case of particle filters you sample the prior distribution, simulate the non-linear plant with these initial conditions and then construct the posterior distribution by incorporating the measurements. This process is computationally more expensive than classical Kalman filtering but is more accurate for non-linear plants.