The limitations, in my opinion, in a neat little concise list are:
1) power - built in flashes lack the power than external ones, they can't throw as far and spread as wide.
2) direction - built in flashes only throw light in the direction of the lens, they're not on a swivle like most external flashes, and are typically used by bouncing light of cielings and walls.
3) Wireless - a lot of external flashes have the capability to go wireless, which can have some realy nice creative effects, not to mention the ability to accessories further with umbrellas, ring flashes, soft boxes, etc...
Having said that, when using an off camera flash, I typically will use my on camera flash as a fill light, thought I always use a Gary Fog Puffer on it to widen and soften the light.
I'm not quite sure what you mean when you say on aperture prioirty, the shutter speed is not flexible. Do you mean you can't adjust it? Because you can. BUT, what is flashed (usually the foreground) is not affected by shutter speed since the flash is always faster than the shutter speed. The backround, however, is effected by the shutter speed as per usual.