Once a character successfully starts grappling a target, is there now any difference between the aggressor and the defender? Or do both characters get exactly the same mechanical effects, such as no dex bonus to AC, no threatened squares, etc.?
The various rules of the grapple maneuver repeatedly refer to whether or not you are grappling, not who grappled who:
If you succeed, you and your target are now grappling [...]
While you’re grappling, [...]
[...] while grappling
Perhaps most tellingly:
When you are grappling (regardless of who started the grapple), [...]
There are exceptions, however. Improved Grab, for instance, has the following:
The creature has the option to [...] simply use the part of its body it used in the improved grab to hold the opponent. If it chooses to do [this], it takes a −20 penalty on grapple checks, but is not considered grappled itself; the creature does not lose its Dexterity bonus to AC, still threatens an area, and can use its remaining attacks against other opponents.