Thanks for that link JP. Interesting, not to mention scary!

I seem to remember being told it was only the current too.

However, ultimately, we are just a resistor (but not necessarily constant), and, while the current through a resistor is proportional to the voltage, the power dissipated by a resistor is proportional to the square of the voltage. Also, each cell in our body is just a resistor, so the power is ultimately being dissipated in the cells that carry the current.

I suspect that the power dissipation only results in some additional heat that the cells are able to deal with if the shock is of short duration, whereas the current completely disrupts their normal operation, but I really have no idea if that's true or not.