Admittedly, my personal definition isn't particularly rigorous. An invention or achievement is useful if it makes other people more able to accomplish their existing goals, or maybe if it gives them something to do when they'd otherwise be bored. It's interesting (but not necessarily useful) if it makes people happy, is regarded as having artistic value, etc.

Relevant examples: Emperor Norton's peaceful dispersal of a race riot was useful. His proposal to construct a suspension bridge across San Francisco Bay would have been useful, had it been carried out. Sylvia Plath's work is less obviously useful, but definitely interesting.