Definition of difference between compression and filtering
My personal view: it depends a bit whom you talk to.

In the SP community, if people talk about "filters", more than 90% of the time, they will refer to linear filtering, in particular filters that can be described by $n$-th order linear differential equations, leading to rational functions in the Laplace/z-Domain. For such filters, we have an enormeous amount of tools to describe their input/output behavior and use this to design the filter coefficients with respect to all kinds of criteria.

On the other hand, when people talk about compression, it will in general involve nonlinear methods, since a purely linear compression is just not that efficient (sidenote: compressed sensing does exactly that, but it is not a compression method really, it's about the efficient sampling of analog signals and yet often mistaken for a compression method). After all, we would expect two very similar input signals to be compressed to the same representation and this already means there is some nonlinearity involved.

That said, some filters that are very commonly used are nonlinear, like the median filter, and there is no real reason to limit filters to linear ones. One could consider a much wider range of nonlinear filters. It's just in general harder to design these, at least from a classical filter design point of view. 

You might still refer to a compression scheme as a filter and some people in SP will do that. I'd say it's a bit less common but that's just my point of view that is also heavily biased by my background.

Summarizing, also in response to your comments: 


In compression, the goal is to represent a given signal with a minimum amount of data. If digital data is already available, the goal is thus typically to reduce the amount of data. 
In filtering, the goal is typically to extract certain parts of the signal by suppressing others. This means that the amount of data may not change at all in a filtering process. Yet the term is so general that in some cases it might.
Compression will apply some sort of filtering (and maybe more).
The terms area not used consistently in the SP community.