I made this same mistake (in a different scenario) just the other day. You actually prove my point, since the max current from a single cell will be the max current of the battery.
The difference is voltage and power: the single AAAA will have 937.5 mWh, whereas the 9v battery will have 5625 mWh.
As I posted earlier, once upon a time I read this and found it to be the case. Things may have changed. I wish I had a 9V here, just so I could see.
-j