Both satisfy $\rm\:f(n+1)-f(n) = (n+1)(n+1)!,\ f(0) = 0,\:$ so they are equal by the uniqueness theorem for first-order recurrences, a.k.a. the fundamental theorem of difference calculus (which has a completely trivial inductive proof).
As I often emphasize, uniqueness theorems provide powerful tools for proving equalities. See also my many posts on telescopy.
On the general topic of functions defined by mathematical induction ("inductive or recursive definition"), see the award-winning Monthly exposition by Leon Henkin, On mathematical induction, AMM, 1960.
Remark $\ $ Note that some of the other answers simply give the standard inductive proof, using ellipses rather than a formal inductive argument. This is most certainly not a "proof without using mathematical induction". These proofs are all standard mechanical applications of telescopy.
A proof that a statement is true for all integers must - at some point or another - employ mathematical induction. The use of induction may not be obvious - it may be hidden (far) down the inference chain in some other theorem or lemma invoked, as in said uniqueness theorem for recurrences (difference equations).
The idea of using uniqueness theorems certainly does differ from the standard inductive proof. Indeed, the uniqueness theorem for higher-order recurrences may be viewed as a generalization of the vivid telescopic cancellation that occurs in the first-order case. One can find some examples in the linked posts.