I apologize if by trying to clarify the ongoing discussion I repeat something already well understood and accepted concepts. We very often refer to them as 'systems' like classification systems or subject headings systems. The difference between these is not in the display, as thesauri when properly designed will have both an alphabetical listing and a systematic or classified display. Furthermore every bibliographic classification has established relations between concepts so relationship between concepts is not particular to thesauri only. Also, every bibliographic classification provides some sort of alphabetical subject index, and sometimes even a thesaurus to enable concept location.
Classification is, it appears, more confusing for non-librarians.
There are many classification schemes and there are also many classifications of knowledge which can be confused with bibliographic or documentary classifications as they can both cover a specific field of knowledge or the whole system of knowledge. They are usually able to express not just knowledge, but the form in which that knowledge is recorded, the language in which it is presented and many features particular to the instantiation of the particular subject within some document like object. ARE NOT and CAN NOT be taxonomic. This is of critical importance for information retrieval as aspect classification helps to establish the context in which one concept or phenomenon might be studied within the document. These systems give provision to describe not only subject, but the form in which it is presented, the time and place that subject is connected, the language it is presented in the document, the physical quality of the carrier etc.