All good questions.
The statement about lattice spacing being similar in both materials is more of a practical issue. If the lattice spacing is not similar, then it is not possible for the two materials to adhere to each other without a huge number of defects forming at the boundary ("dangling bonds", missing atoms, etc etc). In such a case, these imperfections end up completely dominating the physics. This is why GaAs and AlAs are chosen so frequently -- their lattice structure is almost identical. Here is a gorgeous picture of a nice GaAs/AlAs system
Here's a picture of a Cu/Nb interface (both metals, so not a candidate for a quantum well, but still a nice photo)
In the picture you can see that the interface is imperfect, and there are defects where the arrows are pointing. Such defects could have trapped localized eigenstates, or could be strong potential scatterers if you tried to build a quantum well out of materials with such defects.