There is really no difference. You can frame the Kalman filter as a Bayesian posterior inference problem.

For example, for a stationary linear Gaussian model, you have a transition model of the form: z_t = Az_{t-1} + Bu_t + e where e ~ Gaussian(0,Q) and an observation model of the form: x_t = Cz_{t} + Du_t + d, where, d ~ Gaussian (0,R)

Since, z_t and x_t are both multivariate gaussians in this model, you can compute the posterior distribution on z_t's, which will also be a Gaussian. That is basically the Kalman filter.

As the writeup mentions, you might choose a non-Gaussian noise model, in which case the posterior distribution is not a Gaussian and then you employ something like a unscented Kalman filter or extended Kalman filter.