More often than not we hear the Greek Pantheon referred to as Zeus, Athena, Hephaestus, and Ares by their Greek names instead of Jupiter, Minerva, Vulcan and Mars. Asides from Hercules it usually stays pretty Greek but at this point we really just seem to use their Greek names+Hercules and nobody really seems to notice.
It depends on what circles you move in, really. I rarely hear people refer to "Hercules" except when talking about the 1997 Disney film.
In the not-too-distant past it was standard to use Roman names for all Greek mythological figures: this is why, say, Alexander Pope's Iliad features Jove, Minerva, and Ulysses rather than Zeus, Athena, and Odysseus. That practice has been gradually diluted over time, but it's a half-life kind of thing: most names are more-or-less Greek now, but some Roman forms have persisted to the present. A chapter of Kenneth Graham's The Wind in the Willows (1908) is entitled "The return of Ulysses" (not Odysseus); Agatha Christie's detective Hercule Poirot performed a set of Twelve Labours of Hercules in a book of that title (1947) (not Herakles; of course that would miss the wordplay, anyway). Even in your post, you refer to "Hephaestus", writing the Greek name with Latin spelling, instead of "Hephaistos"; and there are other figures who still regularly go by Latin names in ways that go beyond typographic conventions: "Achilles" is still more common than "Achilleus", and I've never heard anyone referring to the god "Apollon" in English.
In some cases, the Latin names get periodically re-solidified in the popular imagination by high profile media depictions. A decade ago, the Spartan king who died at Thermopylai was known as "Lee-ON-ee-das" in my part of the world (and also in the part of the world that the actor who portrayed him comes from), but I've heard "Lee-on-EYE-das" becoming gradually more common lately. (There's nothing very Greek or Latin about either pronunciation, by the way: it's simply that popular culture is changing the names, for whatever reason.) In Herakles' case, the Disney film has had a similar effect, cementing him as "Hercules" for people who were born in the 1980s and 1990s. (I may be misremembering, but I have a vague recollection that Disney was even planning on releasing the film in Greece under the same title until popular backlash changed their minds -- I may have muddled that in my head with some other backlash, though.)
Now, aside from these quirks of arbitrary tradition reinforced by popular culture, there are some circumstances where it's genuinely important to draw a distinction between Herakles and Hercules. Of course Herakles was an important figure in Greek myth; but Hercules was fairly important in specifically Roman myths too. This is because some figures of Greek myth were incorporated into the Etrusco-Latin imagination from a very early date, thanks to Greeks settling and trading in Italy from the 8th century BCE onwards. Figures like Hercle/Hercules, Uthuste/Ulixes (Odysseus), the Dioscuri (Dioskouroi), and Aineias came to have important roles in the native Italian mythologies, and were later recombined with their Greek counterparts. I notice that Wikipedia has separate entries for "Heracles" and "Hercules" (though, bizarrely, the "Hercules" article still focuses on Greek stories about him and ignores the Roman ones edit: just mentions the Cacus story in passing).
In Hercules' case, the two most important Roman stories would be the story of Hercules and Cacus (a villain that Hercules, on his way home from killing Geryon in the far west, encountered in Italy and killed); and the story of Acca Larentia, a prostitute who was given to Hercules (the god, i.e. after his death and deification), and who was later reimagined as the mother of Romulus and Remus.
I'm not sure you fully understand the difference between the Roman and Greek Pantheon.
Think of it more as an adaption and an assimilation rather than just just renaming. The Roman's respected the Greek's and traced their roots to common ancestors and drew analogies between their religion and the Greeks but this all fitted inside the Roman religion. Much in the same way that a modern Christian has reinterpreted god from the Jewish faith, one could not say that the god of Chritianity is just the renamed Jewish god. For example you don't find the Lares or Penates in Greek mythology.
It is very bad practice to substitute Zeus for Jupiter, the only place it is appropriate is in a situation where one cannot tell the difference. For example a statue from the Roman period found in a villa on the Greek mainland.
I would hazard that Hercules became the more commonly used name because throughout the medieval and renaissance period Hercules was still a popular figure and as the bible was in Latin people were reading the Latin myths rather than the Greek ones. This meant most medieval allegorical and mythological texts were written in Latin.