Where do the weight savings come from? Hydraulic fluid is heavy; it seems like you could save quite a bit of weight by carrying less of it if it wasn't expendable. Not that I don't believe it, just curious.

Seems like they actually use RP-1 (rocket fuel) as the hydraulic fluid so that it is burned as fuel after it has been used. This means that the hydraulic fluid isnt "dead mass"

Pumps and pipes. With an open system, you don't need to recompress the fluid and move it back, and from what I read, that hardware weights a lot.

Correct! Instead of using compressor(s) and recirculation plumbing, you're using an accumulator [1]. Weight and complexity savings.

[1] http://en.wikipedia.org/wiki/Hydraulic_accumulator

Not to mention the power supply - what's going to run the pumps when the engines are shut off?

I'm guessing you'd need more pistons, pumps, piping, etc. in a closed system. Probably need to repressurize the fluid.