Sound like the same to me, can any one explain?

Mature is the gore, drugs, torture, etc and I guess reasonable sex scenes tag. Nana has sex and nudity (+ drugs/child prostitution/porn acting) but one to one equivalents in TV-Ma/Rated-R productions exist. Dorohedoro is another good example of just Mature. There is a lot of nudity but it´s not sexual and the gore clearly sticks out.

Adult is the pure tag for explicit nudity/sex. Basically an NC-17. It also goes with hentai.
You can apply both if both fit. Battle Royale, Gantz and Wolf Guy are the poster kids here.

Don´t throw Ecchi on manga with rape, gore or hardcore sex scenes and Smut is the female equivalent of it. Meaning that these further need the proper gender demographics to be used correctly.

I also read EU/US comics and am a librarian.
Manga-Masters, My ANN-Lists + Imdb