The ergonomics and weight balance of a camera are more important than that of a flash. By incorporating a proprietary battery into the camera design the manufacturer has more flexibility on its size, shape and position in the camera. This can reduce the size of the battery compartment, simplify the access door and battery contacts, and means that more power can be stored in a tighter space. This is particularly important in compact cameras. I don't think it's just about the power output and capacity of the batteries. I imagine a more holistic approach to the design of the battery and how it integrates with the camera is involved.

Also, considering that the voltage options for off-the-shelf batteries tend to be nominally 1.5V per cell with the exception of a PP9 (9V), more than one cell would be required to deliver sufficient current to operate the mirror, shutter and lens effectively. In an SLR I expect you would need at least 4 or 6 AA batteries to achieve that, adding considerable bulk to the camera and your camera bag (imagine the spares). :)

Finally, let's not forget that the flash contains a capacitor because of the high voltages involved and that power is transferred from the battery to the capacitor prior to firing -- in this respect, the battery isn't directly driving the flash, the capacitor is, so the battery specifications are less important in the design.