You may not have to reformat either, just plug it in and see if it works.
Of course, if you do re-format, you will lose all the data on the drive, so make sure you have everything that you want already migrated to the new Mac.
On thing that is sensational about Macs is Time Machine (not to be confused with Time Capsule). Time Machine is an automatic backup program that works in the background and is basically invisible to you after the initial full backup. It runs every hours and only writes to the disc what has changed. Could be hundreds of files of multiple GBs of data (like if you are ripping DVDs as an example), could be absolutely nada, zip, zero K (like overnight when the machine isn't on).
Not only is it a Godsend for a full-on HDD crash, but it's terrific for those moments when you accidentally delete a file or save over a file and want to revert to the version you had this morning. The interface makes recovery any file a breeze.
Your old backup drive will be perfect for this. Although, depending on its age and size you might want to think about spending around $100 and getting a new drive. I upgrade my backup drives about every 18 months and current externals are running for approximately $60 a TB...2TB drives typically come in on sale for just over $100 these days.
Apple recommends a drive at least 1.5 times the size of your internal to allow a backup of all contents PLUS multiple versions of your drive's contents going back a year or so. (Since each backup is really small each hour, being able to go back to what files you had on your system a year ago really isn't as amazing as it sounds. )
