It’s certainly not just a fancy name for an electric motor, but the term “digital” was added by Dyson (I believe) to make his vacuum cleaners sound more exciting. Before that they were known as “brush-less” DC motors.
In simplistic terms, an electric motor works by having a moving electromagnet attracted to a fixed magnet, with the electromagnet on the spinning bit in the middle. Turn on the current and the electromagnet gets pulled towards the fixed magnet. Yeah! We have motion. However, when the two magnets reach each other they’re going to “stick”, so at that point you need to reverse the polarity of the current in the electromagnet so it changes from North to South, and it will then push itself away from the fixed magnet – carry on spinning.
In a traditional motor this is achieved using a commutator, which contains two contacts (+ve and -ve) called “brushes” which wipe against contacts on the spinning bit and provide power to the electromagnet. As the motor turns 180 degrees, the contacts are arranged so that the polarity is reversed – pull becomes push and vice versa, and the motor continues to rotate another 180 degrees – where it repeats.
To run smoothly you need more than two magnets, so something is always pulling, but that’s the general idea.
These brushes are a PITA. They cause friction, wear away, and make sparks. But it’s 19th Century technology and the best we had for a long time.
A brush-less DC motor uses electronics to switch the polarity in the electromagnet. It uses a sensor on the spinning bit to work out which electromagnets need current and when. This can be done with analogue or digital electronics according to taste. Hence someone decided to call it “digital” after a while, because digital was “cool”. However, a digital motor has many advantages – not least of which is getting rid of the hated brushes and the need to replace them periodically. If you have smart electronics the motor can be more efficient by applying exactly the right current for requirements, to more electromagnets. It’s similar in a way to electronic fuel injection motor vehicles (if you’re old enough to remember carburettors) – there’s a chance to put in the right power at the right time, saving energy and improving performance.
An AC motor doesn’t have the commutator problem, because the AC current is changing polarity all the time anyway. You could look at a brush-less DC motor as a AC motor, with the DC converted to AC by electronics.