What is the difference between using backup software and copying?
There's a difference between backing up files, and making an image of a disk. Perhaps this is the difference you are getting at.
If I make a backup of my entire disk, by copying all the files on it, there are a few things that I am missing.
1) Filesystem metadata. If I copy files from an HFS volume to a FAT32 or NTFS volume, I lose the resource forks from the filesystem
2) Any boot information. Once I have copied the files from volume A to volume B, I do not have any of the boot information associated with the original volume. So, I cant take that disk, and use it to start up my computer, it just contains a copy of every file that was on the original disk when I made the copy.

If you need copies of your documents, then you need backup software. If you need a disk image that you can boot from then you need imaging software.

Some backup tools do both. The classic unix dd makes a bit-for-bit copy of a disk, and you can deal with the output from it in a number of ways, making it a disk archive swiss army knife.

dd is a tool that requires that you know all about the source and destination filesystems including things like block sizes and such. It is not for the faint of heart.

I've never tried, but you might be able to use th Mac OS Disk Utility to create a dmg image, which is something like what you are looking for. Another thing to look at is time machine. Time Machine creates a Sparse Image on volume, it then creates 'bands' (I think that is what they are called) that store the differences in each file from one run to the next.

You could use a SparseImage to create images of your disk, which you could then make boot disks from.

This is a whole lot of not very accurate technical detail crammed into a small answer. I hope it helps.