Roughly, the HOMO level is to organic semiconductors what the valence band maximum is to inorganic semiconductors and quantum dots. The same analogy exists between the LUMO level and the conduction band minimum.
Why such distinction even exists? If it's the same physical concept, why use different names?
Because convention. Yes, that’s not the answer you wished for but it’s the way it is.
Coming from an organic perspective, one would think of concepts such as linear combination of atomic orbitals or others to arrive at molecular orbitals for entire molecules. Since organic chemistry always deals with rather small molecules and since these have discreet energy levels with discreet differences, it makes sense to speak of a single HOMO, i.e. one single highest occupied orbital (and likewise for a LUMO). The larger a molecule gets, the more orbitals it has and the less the difference between HOMO and LUMO might turn out to be, but from an organic perspective it is still a molecule and must begin and end somewhere.
(Note that this applies to a lesser extent to polymers and carbon nanotubes etc. This is where the shady grey area in which unicorns live begins.)
From an inorganic perspective, if you are dealing with solid materials, they will usually be crystalline in some way; hence they have infinite extents by definition. It no longer makes sense to think of discreet orbitals but rather one realises that certain levels contain a large number of orbitals — hence bands.
If organic molecules end up being sufficiently large, there is no longer a relevant physical difference between HOMO/LUMO designations or bands. But because nobody likes throwing their traditions and customs overboard, the two different names exist.
