Why do we need both Hamiltonian and Hilbert Space to specify a Quantum System?

There is a mathematical reason and a (sort of) physical reason:

Mathematical reason: The Hamiltonian is an operator on a Hilbert space to begin with. Without knowing a Hilbert space, it doesn't even make sense to speak of an operator on it.

Physical reason (sort of): What looks like the same Hamiltonian, e.g. the free particle Hamiltonian $H=\frac{P^2}{2m}$, can be meaningfully interpreted in various Hilbert spaces. For example, the particle could be moving in an unbounded space $\mathbb{R}^n$ or a bounded space like the torus ($\mathbb{R}^n/\mathbb{Z}^n$). This makes an observable difference: The eigenenergies of the Hamiltonian will be different in the two cases (the spectrum is continuous for $\mathbb{R}^n$ and discrete for the torus).

(The reason I say the second issue is "sort of" a physical issue is that the mathematical reason given above actually eliminates this problem: To be rigorous you should always specify first a Hilbert space and then a Hamiltonian, and you will never run into ambiguities like whether the spectrum is discrete or continuous.)