Recursion and recurrence have more than one thing in common. To begin with, they share a linguistic origin. The term "recurrent" comes directly from the Latin recurrere, meaning "to run back" or "to go back." Historically, proof by induction dates back to Pascal (see the article Blaise Pascal takes on induction), and perhaps even to the Italian mathematician Francisco Maurolico (1495–1575), who stated and applied the principle of mathematical induction in his Arithmeticorum libre. But it was Poincaré who championed it in modern times, going so far as to regard proof by induction as "mathematical reasoning par excellence".
Coming full circle
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In everyday usage (according to the Hachette dictionary), recurrence is the characteristic of something that repeats. The term "recursion" derives from the adjective "recursive," a linguistic borrowing from the English recursive, itself formed from the Latin verb recurrere. For linguists, recursion applies to any process that can be repeated infinitely many times (Hachette dictionary). It is used mainly in computer science, but also occurs in art, biology and linguistics.
Although the term "recurrence" occurs chiefly in mathematics, mathematicians also use recursive reasoning. The construction of the set of natural numbers by the mathematician and linguist (no coincidence there) Giuseppe Peano is a fine example, since is constructed from 0 using the "successor" operation. For Peano, zero exists and is an integer; and if p is any integer, then the successor of p is also an integer. This reflects the term’s everyday definition.
Finally, the American mathematician Stephen Cole Kleene (1909–1994) gave meaning to recursively defined sets by building on the principle of recursively defined sequences. And so we come full circle.
