On the computer that still recognizes it, get all of your data off of it NOW. Now is one of those times where you need to do a full and complete backup before proceeding.
Drivers are not going to affect something in this manner.
You have a failing drive, and something in your friends computer (a stronger 3v3 Line, etc) is causing it to be able to attach. Do not trust this drive for anything important any longer. Get everything off of it immediately.
In fact, prioritize your data extraction. Start with the most important things to you; because the drive could die at any moment. It could also die 2 years from now. But this is almost surely signs of a hardware failure, and you need to warranty it as soon as you get your data off.