A core competence is what you are particularly good at. A critical success factor is where you must do well if you are to succeed.
They are linked but might be slightly different. As you say, a core competence should normally be reflected in CSFs. Often a core competence divides into several CSFs so that a core competence of a cut price airline is to get as much flying as possible from each aircraft. To do that it will concentrate on turn-round times, route planning, maintenance schedules. These are likely to be CSFs.