The main difference is the way information is saved and returned to your system, or better said how it can be saved and returned.
Backup normally lets you choose what to backup, you must not want to backup everything. It might be you just want to backup your own files and folders but not the installed applications and so on. Same thing when restoring backup copies of your files; you can choose what to restore.
A system image is a complete copy of everything on your system drive, and could also include other drives on your system. It stores absolutely everything, from boot settings and files to your desktop settings and favourite folders. When (hopefully never!) your system crashes, computer won't boot any more, you can in most cases restore the image. This restores your computer to exactly the same state than when the image was created. Every smallest detail is restored; apps as you had them installed, emails stored by Outlook or other email client, all your documents, pics and music, in fact absolutely everything you had stored on your system.
I use system image backup exclusively. If I need to restore individual files, I can attach the image as virtual disk and extract files I need, as described in these two tutorials: