The term Gothic comes from the name of the architectural style, which itself was a misnomer. The pointed-arch style which is called Gothic was named so after Neo-Classical styles became fashionable, where the pointed arch style was misattributed to the historical Germanic tribes, whereas it mostly originated in France. The application of the term to things that were a specific kind of 'spooky' came a bit later, with late Romanticism bringing about a certain sort of horror novel, where an atmospheric setting was an essential component as much as supernatural elements, and the setting was often something like an old castle or abbey or such building, and often those buildings were Gothic - because by the 18thC those buildings were already centuries old and a good few had plenty of myths and legends already attached to them. That genre of horror, at that time primarily literary, but with a few illustrations and artworks beginning to emerge, which became the 'Gothic novel'.
As time progressed, the elements of the Gothic novel - the conflict between good and evil, the idea of there being a secret or mystery to uncover, the supernatural entities such as ghosts, vampires and werewolves (or the 'explained supernatural' of the likes of the works of Ann Radcliffe), and the atmospheric setting - got applied to more works in other mediums, such as paintings, poems, plays, music and into the 20thC cinema.
It is this Gothic genre which underpins a lot of the Goth subculture - it has given us a fashion aesthetic, an attitude of dark Romanticism, and plenty of subject matter for songs - however, it has also influenced aspects of metal, and there's plenty of people who enjoy aspects of the Gothic without being members of any subculture too, they just read a lot of Gothic novels and watch vampire movies. Those elements have permeated pop-culture, as well as subcultures, and are very prevalent. Often, when something is misidentified as 'Goth', what has been identified has been an element of the Gothic.
So, what makes Goth different to the Gothic?
I would say - in general terms, with exceptions - that all Goths are Gothic, not all Gothic things (or people) are Goth. The Gothic is the fertile ground from which Goth grew, but Goth is more than just an appreciation for the Gothic, and it has a very specific manifestation.
For example, while I'm a Goth because I like Goth music, try and make it to gigs when I can, participate in the club scene (as dwindling as it is where I live), have an attitude with roots in those movements outside of the Gothic that influenced Goth, and I take aesthetic cues from Goth fashion as well as Romantic fashion, I'm fundamentally also Gothic because that is the underpinning mind set - which, in me personally manifests itself in a love for the Gothic as originated in Gothic novels; the ruined architecture, the spooky castles, the ghosts and vampires, the cemeteries, the ancient curses and that dramatic, Byronic sort of decadence... 'Gothic' is the broader umbrella under which Goth shelters. It's also probably the term which should replace 'Goth' in terms like 'Pastel-Goth' or 'Cholo-Goth' which seem to have very little to do with the Goth subculture, but do have a connection to the darker things in life and, at least with Pastel Goth, I've seen Gothic elements like vampires, ghosts, bats and zombies as motifs.
I've seen terms like 'Darksider' and 'Schwarz Szene' to describe the broader miasma of dark subcultures (I'd say 'cloud' but we're being spooky here!) including Industrial, Cyber-Goth, and some parts of Metal, and I don't think Gothic can really replace that, because certainly some of the darkness embraced in these subcultures is from quite a different angle than that within the Gothic genre - lots of dystopian, science-fiction inspiration exists in Industrial and Cyber-Goth especially, and again, the sort of horror that explores themes of madness, psychopathy, and a generally more 'clinical' (for lack of a better word) look into why people do terrible things, rather than a philosophical or supernatural/religious angle. There are still elements of the Gothic, however - songs about demons, devils, vampires and ghosts, supernatural motifs and a sort of Gothic-meets-science-fiction reminiscent of things like the 'Doom' gaming franchise, it just seems to be less of a thematic core, and certainly less of a strong underlying principle or aesthetic.
[An aside on Pastel-Goth: I've seen the term 'Creepy-Cute' used for what is known as 'Pastel-Goth'and I know that this is probably related to the play on words in Japanese between 'kawaii' (cute) and 'kowoi' (scary), so maybe that is a better term altogether? I also don't know what this aesthetic is called in Japan, which appears to be where it originated.]
I think the word "Gothic" as a term to describe a person whose personality includes that sort of mind-set, those interests, that sort of aesthetic, is under-utilised, and is the answer for a lot of people who aren't Goth specifically, but want an umbrella to identify with - 'Gothic' is probably the better term. Certainly, it's an adjective rather than a noun, and lot of us are used to rolling our eyes (perhaps discreetly) when asked "so, are you a 'goffik'?" but I think it's the perfect term for people who love spooky things, but aren't so keen on the sound of the genres considered 'Goth' and would rather listen to say, metal or darker classical works, or maybe darker folk or similar, and just can't get into fishnet and backcombed hair, and hate clubbing, and as all Goths are inherently a bit Gothic anyway, we're all under the same spooky umbrella, so it's not excluding anyone by saying they're not a Goth, rather including a wider range of people by saying 'we're all Gothic', and most of the arguments over the term Goth don't really stem form what Goth is or isn't, but whether someone is included or excluded from the category.