^13 is just a different way to express ^p in the F&R box. If Use Wildcards is ticked, you can't search for pilcrows using ^p, you have to use ^13. ^13 is what Word considers to be a special character.
Note that you can only use ^13 in the "Find what" box. You must always use ^p in the "Replace with" box.
In other words, ^13 and ^p are not exactly the same.
This is documented in the article at https://wordmvp.com/FAQs/General/UsingWildcards.htm: "You may wish to identify a character string by means of a paragraph mark ¶. The normal search string for this would be ^p. ^p does not work in wildcard search strings! It must, however, be used in replace strings, but when searching, you must use the substitute code ^13."