Tadelakt and Box are totally different leathers. Tadelakt was developed partially to have some similarity to Barenia, which is why it was made in Fauve so often.
Box is thicker and much more finished with a high gloss. Tadelakt is softer, more matte, and has a natural very slight unevenness of color that looks kind of like ripples (thus the name). I have a Tadelakt So Flash Kelly that is always mistaken for Barenia, even by H SAs.
IMO they are both fairly delicate - Box scratches more easily, Tadelakt "bruises" more easily. They are both lovely.
"Matte Box" usually refers to Chamonix, although in current production you could also say Sombrero is somewhat like a matte version of Box. However, all still different leathers.