MacBook – just swap the hard drives of two macbook pros (different years)

You can check out the directions on iFixit here.
There are a lot of steps, but none of it is really rocket science. They explain it in detail.

My bigger question would be whether the two boards are compatible.
Based on the specs for each one (2,1 / 3,1), the main differences are (respectively):

• Processor: TS7400 / TS7500
  
• Video Card (integrated): Intel GMA 950 / IntelGMA X3100
  
• VRAM: 64MB / 144MB

I'm not sure how these effect logic board compatibility, but I'd be careful.

Also worth noting that the white 3,1 actually supports 6GB of memory (unofficially). Get it here.

Hope this helps.

Could you swap hdd cable and see If that help ? Also take a look here, where the general consensus seems to be that the issue is the hard drive cable: https://discussions.apple.com/thread/2648100?tstart=0