Editors of Wikipedia seem to think the differentiation is important. On feral cats Wikipedia states "[Feral cats] are not to be confused with wild cats or with stray cats (alley cats)." The attitude expressed on horses is somewhat more measured but still takes care to make a distinction: "A feral horse is a free-roaming horse of domesticated ancestry. As such, a feral horse is not a wild animal in the sense of an animal without domesticated ancestors."
In contrast the entry on boars/pigs struggles with the differentiation: "Domestic pigs quite readily become feral, and feral populations often revert to a similar appearance to wild boar; they can then be difficult to distinguish from natural or introduced true wild boar (with which they also readily interbreed). The characterization of populations as feral pig, escaped domestic pig or wild boar is usually decided by where the animals are encountered and what is known of their history."
It could be that the differentiation merely provides additional historical information about the origins of particular organisms. In this view it is not a value statement about the existential appropriateness of specific organisms in particular biotopes. However, if this were true, all entries about feral animals would probably only be footnotes in the wiki article that describes the animals' taxa. Whereas feral pigs is just part of the article about wild boars, feral cats have their own entire entry. Clearly, the distinction is more than a mere technicality.
In distinguishing between "wild" and "feral", we emphasis our impact on nature. "Wild" represents a state of the world untouched by humans, whereas "feral" represents a state of chaos caused by glitches in our efforts to tame our environment. Number one, I believe life is far more adaptive than we think. Number two, we are part of nature and should not be thought of as intruders. "Feral" does not mean an organism that is inappropriate to its environment. It simply means an organism that is adaptive and can more easily transition between an environment where humans are actively involved and one where humans are absent. Hats off to pigs and cats!