The fully-general continuations that call/cc creates require copying chunks of the stack to the heap for storage and back again when they're used. (Or pointing the stack pointer at the heap, which I find even more cringe-inducing.) They're memory hogs whose lifetimes are hard to reason about and they provide relatively little benefit to most programs, which is why Common Lisp doesn't provide them.
My main interest: delimited continuations are only valid within a specific stack extent so they're much for viable for extensive use - one could argue that async/await is kinda the simplest (and most limited) possible delimited continuation implementation.
They are faster, easier to reason about, less prone to leak memory and are always what you want. Everyone except maybe the chicken scheme people agree, but call/cc is free for them due to their GC strategy.
