(continuation-capture f) --- calls f with the current continuation as argument. In contrast to call/cc, the continuation is not a procedure, but an opaque first-class object. (In Scheme 48, this is called primitive-cwcc.)
(continuation-graft k thunk) --- calls thunk in the continuation k. In contrast to call/cc, k doesn't receive a value computed in another continuation, but rather thunk's effect is performed in k. (Scheme 48: with-continuation.)
The implementation difference to call/cc is minimal, and expressing call/cc in terms of this API is simple:
(define (call/cc f) (continuation-capture (lambda (k) (f (lambda (val) (continuation-graft k (lambda () val)))))))
I can't exactly put my finger on it, but this API seems more powerful than call/cc. In the implementation of delimited dynamic binding in terms of continuation marks I'm working on, there is a need for fetching the marks of another continuation. With call/cc this is impossible, but with this API I can prepend a thunk to the continuation in question, then in this thunk fetch the continuation's marks, and then jump out of it again.