Digital Audio Recorder are the perfect choice if you prefer to record to an actual piece of equipment instead of to your computer, with zero-latency, like in the "old days". Except that, now, digital has replaced tape as the medium. The Digital Multitrack replace the traditional multitrack tape recorders and Digital Handheld models replace the old dictaphones, offering more reliability but still retaining a more familiar feel.
Digital Audio Recorders don't need any software to work, so they don't risk "crashing". They can record straight to their own hard-disks or to SD memory cards. This means that they are a simple, all-in-one solution - no need of computer, audio interface or mic-preamp. Some newer multitrack models even have built-in condenser mics!
If you started recording using tape multitracks, you may feel more comfortable using a digital recorder.
One of the main disadvantages, though, is that they'll offer you a limited number of tracks to record - from as little as just 4-tracks, to as many as 32... whereas if you record to your computer, any software will allow you to use as many tracks as you want.
Digital handheld recorders are still popular, can be used for interviews, college students recording classes or as simple recording devices to capture live gigs and band practice.