First, a little terminology: below the handle is the Ferrule, which is the metal part that holds together all the Bristles. The middle section of the bristles is the Belly, and the very end is called the Tip.
Each of these characteristics is very important in the function of a good brush. If you look at a cheap brush (like an Army Painter, or most of the Citadel brushes), bristles essentially taper from the ferrule to the point. If you look at a Winsor & Newton Series 7, the belly of the brush is quite fat -- wider than the ferule. This allows the brush to hold more paint, so that you don't have to go to the palette as much, and also, the brush doesn't dry out as quickly.
Kolinsky sable is highly sought after for top end brushes because it has a natural tendency to form a tip: just wet a good sable brush, and flick it away from you, and it will form a perfect tip. A cheap Kolinsky sable brush like a Citadel Standard Brush will have a tip that frays (splits up into a fat mess) as you paint, requiring you to re-form the tip, while a S7 will maintain its shape and keep a fine tip as you paint.
As compared to synthetic brushes, sable brushes have pros and cons.
The largest downside to a synthetic brush is that they kink. Every single one will kink as you use it, no matter how nice you are with it. This means that with time, the tip will be bent (no matter what you do), and not form a sharp point, making it not ideal for several types of work, like detail. Synthetic brushes also generally hold less paint than natural hair brushes, although some top end synthetics don't have this problem (they are also just as, or nearly as expensive as top sable brushes).
The largest advantage of a synthetic brush is that they are very springy, and bounce back to their intended shape, pretty much no matter what you do to them. This makes them easier to learn with, because a painter doesn't have to work to keep the brush in the correct shape.
The larger Citadel brushes (standard, basecoat) will not keep their shapes as you work. This means you constantly have to go to the palette, and roll the brush (in the paint) to reform a nice tip. Duncan is a pro at this in the
GW videos
The W&N Series 7 brushes, in comparison, are extremely springy, nearly as much as a synthetic. The Raphael 8404 is slightly less springy, and will tend to stay bent. Technically, this is not a quality issue -- it's because the 8404, by design, has a larger Length Out (length of bristle from ferrule to tip). This allows it to hold more paint, but reduces its ability to hold its shape.
One other advantage of Synthetic brushes is that the material for the better ones (Golden Talkon, I think it's called) is more durable than natural hair. This means you don't have to be as diligent at cleaning it, you can be rougher with it, and, importantly, it wears better when you are using metallic paints, which contain flakes of mica.
Now, your original question included one of price. When I buy them on sale, I can get W&N Series 7 Size 0 and smaller for about $10-$12, and size 1 and 2 for around $17-$20. They are maybe $5-8 more than that when not on sale. This is actually MUCH cheaper than synthetic round brushes over time, because if you care for your W&N S7 (or Raphael) properly, and don't use them for metallic paints, they will last 6 months to a year, *at a minimum*. I have year-old brushes with 1000+ hours that I've given away, that still look new.
In contrast, a synthetic brush that kinks goes straight into graveyard, and that happens in weeks, for me (using brushes about 3 hours a day).
Cleaning natural hair brushes is really important. Basically, after you're done and have cleaned them, they should look exactly like what they did when you opened them new. Over time, the bristles will effectively get shorter, and the tip less sharp, but they should still look awesome. I highly recommend Winsor & Newton Brush Cleaner & Restorer for daily use and Master's Brush Soap and Conditioner for the major cleanings (for me, that's 2-4 weeks).
I also highly recommend keeping a separate set of brushes for metallics (because they will get ravaged :() and one extra brush for super details like eyes, that you don't use for anything else. For the super detail brush, my recommendation is Raphael 8404 size 6/0.
Also: find out who sells Kolinsky brushes in your area -- try fine arts stores -- and ask when they have customer appreciation days and sales. Just buy a couple of brushes when they are on sale, treat them well, and they will last you a long time. They are also much fun to paint with