It's called a cut because, as with b-roll, the film negative (the actual strip of film) was originally cut and attached to another negative so that one image would 'jump' to another. Because of the jarring or unnerving effect they often have, jump cuts can be used to suggest or induce confusion, fear, otherworldliness, or even humour.
Godard’s use of it violated the ‘30-degree rule,’ which states that in order to maintain a flowing narrative and keep the audience involved, the camera must change its angle by at least 30 degrees when filming the same subject. His reason for using the jump cut isn’t entirely clear: whether it was an act of spite towards producer Georges de Beauregard for demanding Godard cut an overly-long film to the agreed length, to cast himself as a visionary director while impressing gullible French film critics, or simply because it was too long and he saw an opportunity to experiment, his breaking of the 30-degree rule pioneered the use of a brilliant editing technique for future filmmakers.
One of the jump cuts early in Godard’s Breathless which made the film, the jump cut, and Godard himself famous.
Unless you’re advertising for a film production company which creates effects like this, it’s probably best to avoid using it in corporate videos. A jump cut is perfect for creating an edgy, unusual sensation in a New Wave film, but would most likely end up as a distraction in a film in which the point is to be as clear and straightforward as possible. Any number of corporate videos from the early 1990’s can attest to that.