By my standards,
Smut & Adult are repetitive, both include a significant amount of sex or extremely close-to-sex fanservice / ecchi moments. Basically choose whichever one you like, or both, and go with it. If I had to try and differentiate between them, I would say that "adult" might be more serious (like an erotica featuring a mature plotline and drama and all that) while "smut" would be less serious and more just fun, but can still involve some plot (like hentai). So, maybe, Adult is "serious sex" and Smut is "fun/fetishized sex." I still think we really only need one or the other, though, as they can very easily fall under each other.
Both Smut and Adult fall under Mature, but not all Mature is Smut nor Adult. Mature, to me, basically just means anything intended for an audience of 18 or older. You can still have deep psychological stuff or things that make you think hard without being aimed at a mature audience. Mature doesn't have to mean something else that is bad, either. To me, it literally just means that your story is intended for 18+ audiences. I would slap it onto all of my stories if I didn't have to be limited in the number of genres I could choose.
Another way of potentially looking at this, now that I'm thinking about it, could be:
Smut = frequent and detailed sex to the point where the sex itself is part of the reason for reading the series.
Mature = "mature themes" that would basically classify any video game as mature, so drug use, large amounts of violence, some heavy sexual elements but without the sex being one of the main selling points, and so on. Anything that would get a game flagged as rated M.
Adult = anything aimed specifically at a target audience that is 18 years of age or older. You can have mature themes without specifically targeting adults. The main difference between mature and adult would be whether you want people under the age of 18 who are also mature reading your story, or if you specifically want to target only those who are 18 or older.
It's all up for interpretation and it can go any sort of way. I want actual definitions to go with every genre and tag so that we have a strict guideline to go by = P
