Filter's which requires physical model
After reading your comments, the real question you're asking is:

"What's the difference between an estimator and filter?"

An estimator is a system that takes in inputs and attempts to estimate a signal (or state) given knowledge of the system. A simple example is system that takes in positional data and calculates the corresponding acceleration. In this case, your input data is position, the system is the second derivative with respect to time, and the output is acceleration. You've just estimated the acceleration by feeding it through a model.

A filter is just a system that takes an input and attempts to modify the frequency content of a signal. A Butterworth filter is a filter, not an estimator. 

So the real question you have to ask your is: "What am I trying to do?"

A problem where you're trying to estimate something, you use an estimator (i.e. a system that produces an output using some physical model). A problem where you trying to change the frequency content of a signal, you use a filter. The filter doesn't care about the underlying process or signal, it's just setup to modify the frequency content of the signal.

Mathematically, a filter and estimator are the same thing (put signal in, get signal out), but they typically have different physical meanings and are used to achieve different things.

Edit: It also seems that your question may have been inspired by the 'Kalman Filter'. Even though this contains the word 'filter', its still considered more of an estimator. However, depending on how you set up the problem (i.e. how much uncertainty you have in your model and how much uncertainty you have in your feedback), the system will experience different frequency response characteristics.