I hope this basic example helps you to see the difference. I'll denote configurations by simple tuples where each number indicates the occupation of a spatial orbital, with increasing energy from left to right.
Let's take only the ground state determinant as reference and do CIS within a space of 4 spatial orbitals (yielding 8 spin orbitals in total) and two electrons. The additional configurations that we obtain when we look at all single excitations are the following:
(2 0 0 0) -> (1 1 0 0) | (1 0 1 0) | (1 0 0 1)
Now let's add a second configuration and do CIS on this new configuration. I'll omit configurations that have already been generated in the former step.
(0 1 1 0) -> (0 2 0 0) | (0 0 2 0) | (0 0 1 1)
You see that you get new determinants based on single excitations starting from this reference. This set of configurations differs from the set that you would have obtained by doing all single and double excitations based on the groundstate configuration. If we had done CISD we would have gotten one more determinant,
(0 0 0 2)
In this basic example the difference is small but as the active space and the number of electrons grows, the number of possible configurations "explodes" in size.
The same transfers to multireference CISD. By adding a reference and doing single and double excitations you will get additional configurations that aren't generated based on a single reference. It is also different from simply doing the next higher excitation scheme, as you can see in my CIS example.
You can also include determinants that correspond to higher excitations by starting from a doubly excited determinant and doing single excitations on top of it, if we had more electrons and a larger active space.
The multireference ansatz allows us thus to include some higher excited configurations without including the whole set of all higher excited determinants, which is often unfeasible due to the large number of possible combinations that are available given a larger active space. It also allows us to add higher excited configurations without the cost of doing the full higher excited scheme.
The diagonalization of the CI matrix based on the generated configurations yields
your electronic states and eigenvalues. The electronic states are mathematically linear combinations of the determinants although many states often have a dominant contribution. This allows us to approximately identify a state with the determinant, that has the highest coefficient in the linear combination.