Is the SCSI Address of that Drive used by any other SCSI Devices in the second computer?
If it is you'll need to alter a Jumper to give it a different SCSI Address and to a Unused one on that SCSI Channel as well.
What Format was on the Drive and what OS was using it in the other computer where it was being used?
What OS is loaded on the computer it is now fitted to and what if any Security did you have in place?
Ideally with any form of Data Recovery you need to understand a few things if this is [b]IMPORTANT[/b] data you should pack up the drive immediately and send it to a Specialist Data Recovery Company and pay them to recover the Data. Also you shouldn't attempt to use the drive for anything as the more you mess with it now unsuccessfully the more expensive the Recovery is going to be and the less likely you are to get it all back. So you'll pay more and possibly get nothing back or only part of the rate recovered.
However if it would just be [i]Nice[/i] to recover this data you can attack the drive with some form of Data Recovery Software. Depending on how much you are willing to pay for this software here is a short list of the commercial ones which seems to work fairly well though I have not used them all.
http://tinyurl.com/273ay5
http://tinyurl.com/rmvrv
http://tinyurl.com/yvrsl
http://tinyurl.com/yyk8cm
http://tinyurl.com/382vm
http://tinyurl.com/ynwnel
http://tinyurl.com/nc4kx
http://tinyurl.com/2wox7l
http://tinyurl.com/374d8b
http://tinyurl.com/djw6u
http://tinyurl.com/f73se
http://tinyurl.com/8bd3f
http://tinyurl.com/afu2j
While the last 2 are from the same company they are very different products. Win Hex should only be considered for use by someone who knows Hex Inside Out while Davory is a great cheap but slow General Purpose Application.
Personally I prefer the On Track Option but it's anything but cheap. However because I use this type of software a lot it's paid for itself many times over by now, for a cheaper option Spin Rite from Gibson Research or Stella have both very good reputations by other TR members.
Then there are the free ones which I have absolutely no experience with but you can try at your own risk
http://tinyurl.com/28a7va7
Col