Have you moved from Windows XP to Windows 8.1? How were you backing up your data? Were you physically copying and pasting files or folders to the drive or were you using some third-party backup software? If the latter, you may need to reinstall the software. How is the external drive formatted? FAT32 or NTFS?
An alternative would be, if your original laptop is still working, is to reformat the Seagate drive with your new laptop (i.e. erase everything), then plug it into your old one, move the files on to the protable HD and then see if the new computer reads it.
Ultimately, if a computer can't read an external drive it is generally a) the hard drive is failing b) the USB cable is faulty or c) the drive has become corrupted.