What is the advantage of having an angled drawing desk?
There are several advantages!
Ergonomics
We'll start with the one you mentioned. If you regularly draw on a horizontal surface, you'll spend a lot of time with your head down. This is terrible for your neck! It can lead to sore, tight muscles. Or, if you have neck problems (like I do), it can be downright painful to even try to draw like that for any length of time.
Perspective
Having the desk at an incline allows you to have a better vantage point of your work. You can more or less have your eyes even with the center of the work piece. This means you're less likely to distort your drawing with perspective problems. When you're head down on a flat surface, your visual focal point is generally aligned to a single spot.
To understand what I mean, take a look at this sidewalk chalk art.
This is how it actually looks:

This is how it looks from a specific vantage point:

In the case of drawing on a flat plane, you may end up drawing a distorted, stretched out image, but your mind perceives it as perfectly proportional.
You can avoid that by using measurements and taking breaks to look at the piece from different angles, but it's a little more work.
Drawing grip
To me, this can be the biggest advantage.
Having an inclined plane allows you to use an underhand or paint brush grip with ease. These grips allow you to control your pencil by using your shoulder and elbow, instead of your wrist. Long, even, consistent lines are possible when using your whole arm. These grips, and your lines, will have a more natural and relaxed feel.
However, these grips aren't necessarily a benefit for all types of drawing. Generally, I'll use them for life drawings and similar work, but not for small character sketches. I still use an inclined drafting desk or tabletop easel whenever I can, though.