From a practical standpoint, the strategic advantages of maneuvering in three dimensions during a battle are massive—while there is no Stealth in Space, moving in three dimensions will add another axis of movement to any attempt to dodge projected energy weapons at anything beyond point-blank range, thus increasing the chances of a successful (ie. not anticipated by the firer) dodge by 50%. A very important thing which must be pointed out here is that Spaceships can't make use of an atmosphere to help them "steer"—rudders and ailerons, which give aeroplanes their maneuverability, would be completely useless on a spaceship. In reality, thrusters (small mass-reaction engines) can provide this sort of maneuverability and are commonplace on real-life spacecraft, but are rarely shown in media. So, pulling most of the cool moves that make use of three dimensional navigation in the air (like some of the scenes you see in Top Gun, for example) would be usually impossible to do with fictional spacecraft - though a whole new class of different cool maneuvers is opened up, such as having ships not pointing in the direction of flight and skidding about on crazy vectors, maneuvering thrusters ablaze, trying to get a good shot off. Remember how much we hated that fucking spaceship in the classic Asteroids game? Well, try to imagine that on a 3-D interface (in fact, play Frontier: Elite 2 and think yourself lucky if you survive your first "dogfight"). Alternatively, though, you could invent a Reactionless Drive for the ships in your setting and use it to justify all sorts of zany behavior.
One consequence of this is that whenever "intentional" use of the third dimension pops up in a story, it is somewhat less realistic—now, instead of 2-D Space, you have Old School Dogfighting, which treats Space as though it has an atmosphere, and spacecrafts are as maneuverable, if not more so, than aeroplanes. Though admittedly this does look pretty frickin' cool, especially if the CGI department gets to cut loose with their skills. However, almost never do we see two ships just happening to approach each other at different orientations in a normal no-fighting scene.