As far as the SIGNAL PATH is concerned, there is no difference between a 4-pin and a 6-pin FW connector.
The extra 2 pins on the 6-pin are just to carry DC voltage to operate any portable device that is built to run from the computer’s power supply.
Plus, the 6-pin plug is much larger and sturdier, so a lot of pro DV camcorders have a 6-pin FW jack instead of 4-pin just so it will hold up to rugged use better.
To your specific problem…
A. you MIGHT have a “bad” 4-pin cable
B. one of the 4-pin connectors on the camcorder or computer is broken
C. the actual internal FW “board” in either unit is “fried”.
Something to try:
Leave the camcorder connected via the FW cable and leave the camcorder in “VCR” mode.
Quit the software on the computer and then open it again.
Sometimes that will “wake-up” and create a proper “handshake” between units.