3 metre USB cable with portable DVD writer will not work with Dell laptop.

The same laptop and DVD writer will however work with a short 60cm cable.

Any reason why?

Power. Most likely the power isn't enough on the long cable. I use a Y cable to get around this.
The Y cable taps two USB ports for power. It's a common cable, found on Newegg and Amazon for example.

I have tried the same cable and DVD writer with a friends laptop and works fine. Could that still be a power issue with my laptop?

Yes. Not all USB ports are the same in this area. Can confuse the issue.