These differ in at least five ways.
a) Who gets which interview in what situation?
Credible fear interviews are for people who are in expedited removal proceedings, who have not been removed or deported from the U.S. before, and who have asked for asylum OR expressed a fear of returning to their home countries, especially if they are children, for whom their lawyers have requested credible fear hearings.
Reasonable fear interviews are for people who are reinstatement of removal proceedings and who have expressed a fear of returning to their home countries. (They may have asked for asylum, but if they are in reinstatement proceedings they are not eligible for asylum. They may have asked for withholding of removal or deferral of removal/protection under the Convention Against Protection, but it is a rare campesino who will know to make such a specific request.)
b) The standard of proof differs;
c) The timing may be different — in my experience, people in reinstatement of removal proceedings languish for much longer waiting to hear what the asylum officer has decided;
d) The form of relief for which the person is eligible will differ;
e) The possibilities of getting out of detention (via bond, “bajo palabra” or under a person’s own recognizance, or parole) may differ, depending on Department of Homeland Security practices at a given moment;
f) It is more rare for a child to find herself or himself in reinstatement proceedings (and thus in a reasonable fear interview).
See this wonderful chart by Maria Andrade & Associates, outlining the differences as of October 2014.