These two genera appear to many to be the same but have subtle botanical differences, the most obvious to gardeners is that Ophiopogon produce berries while Liriope produce dry seeds (albeit rarely!).
Neither are really known by common names in the UK but the "black grass" Ophiopogon planiscapus 'Nigrescens' is well enough known by that, slightly misleading common name, it is not a grass at all of course!
All of these plants make dense clumps or patches of evergreen foliage and can be useful for edging or groundcover use in shady places. They all tend to do far better if grown in moist shade but will survive in a wide range of sites. Some forms (including the often requested Ophiopogon japonicus) can be rather slow to increase, hence quite expensive to produce. If a good, vigorous, but not invasive, groundcover is required Ophiopogon bodinieri is the better bet, covering the ground well, even in full sun.
The classic black grass. Ideal for tubs and containers as well as edging borders etc. Strappy black foliage and short spikes of mauve flowers followed by black, pearl-like, berries. Decent soil in shade or part shade.