Short answer is that natural bristles tend to be finer and more resilient for applying finishes...but require more careful handling and maintenance. Synthetic fibers tend to be less flexible to at least some degree, but are tougher and less 'fiddly' to clean and maintain.
Synthetic brushes are especially recommended for acrylic type paints. Natural fibers can actually separate components of such paints at a microscopic level, leading to greater likelihood of visible brush-strokes and general 'stickiness' in use.
Natural fibers lend themselves particularly well to oil-based paints...since in their 'original' state, oils of various sorts are actually part of their make-up. Natural bristles are also the preferred type for lacquers. On the whole they will give smoother finishes for those types of paints.
Good brushes of either type will obviously cost more...more important, probably, for natural bristles, where cheap ones will shed hairs at the most inconvenient times. Cheap synthetic brushes won't usually shed individual fibers...instead the whole brush part will often weaken from repeated solvent contact, and separate from the handle.
But there's a place for cheapo brushes as well. For dry-brushing, for instance...which I mainly do with enamels...I like cheap, fairly stiff-bristled synthetic brushes, because they stand up well to the sort of abrasive technique required. They can actually be 'squashed' into different useful shapes, which would simply wreck natural bristle brushes. (It's pretty hard on synthetic ones as well...but being cheap, they're easier to replace periodically.)