GPSGate will maintain a log and let you play it back. There are versions available for both PC and PPC, so you could get one of each and play it back on the PC. It stores the NMEA input, so it looks exactly like a GPS to the software when played back. Any mapping software that has a PC component should work (iGuidance, Routis, etc).
Also, there is a package called VisualGPS that has a verson for both the PC and PPC and does logging. I'm not sure about playback, but you can check at
www.visualgps.net for the details. The PPC version is freeware, IIRC, but the PC version is shareware.
I don't think either will shut off the screen, but you can set it to auto-dim after a few minutes and cut the power demand pretty significantly.
If the Axim battery life ends up too short, then you may want to look at the Delorme BlueLogger GPS. It can store data directly and at one point every 2 seconds it will store about 8 hours of data, maybe more. You can move that to a longer time, say 1 per 10 seconds and get literally days of data. The battery lasts about 12 hours.