It's the application which dictates whether you should use a linearized, extended, unscented Kalman filter or you can go easy with the simpler linear Kalman filter.
In other words, the extended Kalman filter is not preferred over the linear one because it provides better performance than the latter. Just the opposite; its estimation error is larger than what could have been provided by an equivalent linear Kalman filter if the problem could be solved by the linear.
When the physical nature of the problem does not permit a linear mathematical model, then you should either abondon the Kalman filter at all, or you should find a way around (such as the linearization of the nonliner model) to continue using the Kalman filter and still expect its benefits unless you have some alternative algorithm to go with.
Most fundamentally the accuracy of the extended Kalman filter depends on the the sampling interval (step size) used to update the linearized matrices and other parameters. Hence you should use dense enough sampling so that the nonlinearity does not change too much to add excess linear apprximation error. However smaller sampling interval also means more computations per second.
Also subtle issues such as noise de-coupling, independence assumptions, dimension reduction have thier own solution strategies which can also affect the accuracy of the filter estimates per given sampling rate.
When your problems are inherently nonlinear, then you cannot use the linear Kalman filter, therefore you can either use the extended Kalman filter or think about other algorithms that could be useful.