I have owned and obviously used both, many times with different paints... Below are my opinions of pros and cons.
Flatline:
Pros:
-Looks
-Increased Distance
Cons:
-Barrel has a rough powder coating, that works to give the ball traction to actually spin. It works great, but will eventually wear down, decreasing the effectiveness of the Barrel.
-Decreased accuracy for short range shots
-Fixed amount of backspin (where the apex allows you to set desired amount, or turn off completley)
-Loud
-Not compatible with freak system
-Picky on paint
APEX:
Pros:
-Price
-Adjustable amount of backspin (can even turn off completely)
-ability to curve, sink and float shots (although curving is VERY hard to do)
-Able to handle more paints
-Can be fitted to almost any barrel system with the Electrical tape mod.
Cons:
-If paint does break (which is not that often) it is a pain to clean.
-Looks
-Soft rubber paddle that creates the backspin, wears with time and will eventually need to be replaced
Also keep in mind, although both barrels increase distance, unless you are using a very high grade paint, chances are the shots will not break even if you hit your target... usually the paintballs are going so slow they will just bounce.