inetdog: The official explanation is that the Li battery can deliver enough power for several starts of the APU and then be quickly recharged again once the APU or other power source is running. NiCd's higher internal resistance makes very rapid recharging more problematic.
I've been poking around trying to learn which of those "facts" about various batteries are true, without much luck. The wikipedia article is not very satisfying, and I don't know the merits of this page
Lithium-based Batteries Information. The table would have you believe that LiCobalt is a very poor performer compared to later Li technology. It would be interesting to know Boeing's supplier's reason for that choice.
On the different topic of catching the fuselage on fire, early on I posted a link to a 2007 paper produced by U of Maryland (iirc) for the FAA which showed that Boeing's composite fuselage recipe was well-behaved in that regard. Perhaps someone has revisited that effort, given the embarassment with battery engineering.