I mean no disrespect, but this thread leaves a lot to be desired! If I were a new user and wanted to know the difference between a Nandroid backup and using Titanium Backup I would still be lost..
Let me attempt to clarify.
A Nandroid is another term for image. In computing an image is an exact copy of whatever you imaged bit for bit, and said image is stored as a file. Therefore, if you create a nandroid of your phone, you are creating an exact copy of the phone, and that nandroid can be restored at anytime in-case damage incurs at a later date. Its always good practice to keep 2-3 date stamped nandroids as well, and to off load those to a second location like dropbox or google drive.
Creating a Nandroid image is another animal and the prerequisites are: You must have root user access and a custom boot-loader installed. (Many solid write ups on how to do this are stickied!)
(edited)
Titanium Backup handles backing up your apps (and their settings), freezing apps, and uninstalling apps that you don't want (aka bloatware.) The main functionality of Titanium Backup is useful if you have saved games or settings you want to keep while switching between ROMs. However, Google is working to allow app syncs to the cloud, so the need to use Titanium Backup for backing up apps will soon phase out, but it'll still be useful for freezing and removing those nasty preinstalled applications.