Short Answer: It is potentially hard (as
bRost03 indicates in the comments). To be precise, coNP-hard.
In adiabatic quantum computation, the ground-space of the final Hamiltonian is typically determined by the optimum solution to some constraint satisfaction problem (CSP). If the CSP is perfectly solvable, the ground-space is spanned by (standard basis states which encode) the assignments which satisfy the constraints. If the CSP is not perfectly satisfiable, the ground-space is spanned by all assignments which are closest to being satisfying. In this case, your question amounts just to a question of finding out when the set of satisfying assignments — or the set of optimal assignments — of a CSP are the same.
We can reduce any such CSP to a boolean formula. So what we're looking at in the case of asking whether the satisfying set of two CSPs are the same is whether a boolean formula of the form $$ \varphi_1(x) \iff \varphi_2(x) $$
is a tautology, which is to say, whether that biconditional is true for all $x$, for some boolean formulae $\varphi_1$ and $\varphi_2$ provided as input. This makes it a special case of " TAUTOLOGY ", the decision problem of whether a logical formula is a tautology.
The problem TAUTOLOGY is in coNP
— in fact, it is coNP complete, because it is in effect the complementary problem to the NP compete problem SATISFIABILITY. This shows that the special case of determining whether the satisfying solution sets of two CSPs are equivalent is no harder than coNP. But because we can test whether $\varphi_1$ is itself a tautology by setting $\varphi_2$ to the boolean formula 'TRUE', this special case is as hard as TAUTOLOGY is itself, so it is coNP hard.
The more general problem of equivalence of optimal solutions contains this problem as a subcase, so it is also coNP hard.