While approximate computation may be most readily used for processing data for human sensory input, it could also be useful for certain test-and-confirm type problems (where an approximate test--not even necessarily absolutely excluding false negatives--can filter out a majority of uninteresting results [distributed computing projects tend to work this way, the Large Hadron Collider also uses data filtering which _might_ be amenable to approximation if the false negative probability was sufficiently low]).
Something like a search engine could probably use approximate computation (the result is a list sorted by estimated fitness).
It might be possibly to increase the effectiveness of a safety system by allowing the use of more data and more processing even though the processing is approximate. Likewise, a self-correcting system (e.g., a flight control system) could tolerate minor errors.
Much simulation is effectively approximate (e.g., modelling of the gravity of distant objects as a single point object) already. In addition, measurements are inherently approximate and incomplete, so using approximate computation might be less inaccurate than one might naively expect.
Error detection and correction are also probabilistic, and so might be able to use approximate computation (Lyric Semiconductor's ECC product?).