Welcome to CDRLabs, ngonqua!
What interface is the drive (i.e. USB, Parallel, etc.)? Does Windows "see" the drive? What version of Windows do you have on the laptop and desktop?
Edit: did the version of Nero on the desktop computer come with a burner other than your HP? If so, what burner did it come with? The reason I ask this is that "bundled" versions of Nero frequently work only with the brand of burner with which they were included.