Thanks everyone.
Yes I'm using the Y-cord USB that came with the drive, on both computers. So that's not the issue.
I also just plugged the drive into two different USB ports on the rear of my Lenovo and still got a failure.
Granted, one was a USB 2.0 and the other was a 3.0 ... but I just realized the drive itself uses two 2.0 ports, so in theory, that shouldn't matter... right?
Just for good measure, I plugged the drive into the two USB 2.0 ports on the front of the Lenovo, as I'm almost positive they use a different bus off the mobo. (My dogs are being particularly fussy and in my face, so not an idea evening to open the case up). The drive failed on these ports, too.
I went to the Device Manager and right-clicked, "Update Drivers" for every single item under Universal Serial Bus controllers; all said the best drivers were already installed. I had Windows Update do a check; nothing new available. And they all look pretty generic, so I'm not sure I'd know what specific hardware manufacturer and model I'd want to search the web for drivers on. I'll also point out that no other USB drives on this computer have been acting up; occasionally I'll have a WiFi drop, but that's an onboard chip, not coming from a USB dongle.
I'm at a loss here. At some point this weekend I'll burn a few more copies on the HP, to see if the successful burn was just a fluke. I'm about to bite the bullet and just buy a replacement drive, but I don't want to dunk $$$ if 1) it's not going to resolve the issue, and/or 2) there's some kind of software or firmware fix.