As an easy example, for $$n$$ a positive integer the sum of the first $$n$$ positive integers is $$1+2+\ldots + (n-1)+n=\sum_{k=1}^n k$$; however, this can be expressed in a closed-form as $$\frac{n(n+1)}{2}$$, which provides an efficient way to compute particular cases and gives a hint at asymptotic behavior. In a similar vein, $$1^2+2^2+3^2+\ldots+n^2 = \frac{n(n+1)(2n+1)}{6}$$; Eduoard Lucas conjectured that this value is only a perfect square for $$n=1,24$$, an observation difficult to spot without an explicit formula to work with. Other cases include Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$, which expresses an infinite series as a combination of three values of an elementary function (the Riemann zeta function), and Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2$.
That being said, sometimes other properties, such as recurrence, are more helpful for various purposes. For example, starting with $$F_0=0$$, $$F_1=1$$, a closed-form for the Fibonacci numbers is $$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$$ . This gives us the asymptotic $$F_n\approx \frac{1}{\sqrt{5}}\phi^n$$, but to compute $$F_{20}$$ it is much easier to use the recursion $$F_{n}=F_{n-1}+F_{n-2}$$. The recursion is likely more useful if one is doing combinatorics as well.