Two different characteristics, Debbie.

Inlaid means someting has been "laid into" the clay, usually in the center, while the clay is still soft. If the inlay is paper with something printed on it (litho, usually) the chips are usually called "crest and seal". The inlay has to be flush with the surface to meet the Eisenstadt criterion for C&S.

Die-cut refers to one of the kinds of things that can be inlaid.... usually a complex pattern that has been cut from paper or plastic by a metal die in a press. Technically, most all inlays are die-cut, even the round printed inlays, but the usually use of the die cut term is for things like initials and monograms with no printing on them.