The distinction between nihil and nihilum is a very fine one which, no doubt, the Romans learned to apply instinctively. It causes hardly any difficulty in translating from Latin, but in writing idiomatic Latin it does need some care.
Nihil (or nil) is an indeclinable substantive having, very simply and without qualification, the pure sense of 'nothing', or 'not anything'. 'Nothing', referring to nothing else, is absolute and can only have a singular meaning. Nihil isn't used for any other purpose. However, it is equivalent to nulla res, which can be used in those slightly awkward cases where an inflection is necessary because of its position in a sentence — Smith & Hall's 'Copious & Critical Dictionary' gives examples of this, e.g. Nepot. XXI. 2 (end), nullius rei [NOT nihili] _cupidus, nisi imperii, and Cic. Brut. 216, nulla re una magis oratorem commendari quam verborum splendore et copia.
Nihilum, also substantive, conveys a sense of quality, 'nothingness', or 'non-existence'. Like nihil it is necessarily singular, but it can take an inflection acording to use. This explains its use in, for example, conveying the idea of 'worthlessness', as in nihili facere, where it is a genitive of value (sc. 'of nothing at all'). In the ablative it occurs as ex nihilo, 'from/out of nothing' ; in familiar adverbial usages such as nihilominus, nihilo magis ; and to indicate value, as in non nihilo aestimare ('not worth nothing', 'at any rate,worth something' etc.).