1. more precise: you control the angle of the blade & the closeness of the shave.
You can actually see where you're cutting, which means shaving is simpler, and shaping stubble is much easier. It also means you don't pass over the same patches several times just to get a pesky hair or two. Fewer passes makes for less irritation and better skin.
2. Cleaner: a single blade cuts through even thick stubble more easily and clogs less.
That means if you're a hairy chap looking for a nice clean shave, or if you're like Antonio (one of the co-founders of Thomas Clipper) and often have a couple days stubble to get rid of when you shave, then double edge shaving will be a dramatically smoother experience.
3. better for Sensitive skin: less tugging means a smoother shave.
Every extra blade you add to your shave, whether electric or wet, means more pulling and tugging at your hair follicles. This can cause razor rash. Many people (including Matt, one of the co-founders of Thomas Clipper) find that double edge razors dramatically reduces this, leaving them with smoother skin after shaving.
A skill worth acquiring...
At the end of the day, shaving with a double edge razor is different. Like anything worth doing, it takes a little time to get the hang of (we think around 3 shaves).
But once you've mastered DE shaving, you'll never go back to plastic. On top of that the blades are a fraction of the cost. Which is a nice bonus in an altogether better shave.