Usually what I do is to save multiple copies with incremental numbers, but I find that this isn't a good method because I end up with a folder full of word documents of the same file + changes. It isn't really easy to organise because of it.
In a version control system you can perform such things as
diffs to basically get a view showing changes from one version to another. You can also create branches, merge branches, and so on. Depending on where you host and what you use, you could also have comments at certain lines (all the way to the extent of discussions). This is usually outside the realm of the source version control, and usually in the webview.
The advantages go beyond single user use. You could for example have a proof read copy and do a diff to check what the person changed. Similarly you could coordinate work with other people responsible for certain aspects of how your writing is suppose to look, proof reading, editing, and so forth, and all this with out going though a iterative time consuming "pass me the last copy" process.
I personally recommend something like
git for the simple reason that unlike subversion you can work completely locally anywhere, anytime ie. if it's not too clear, with out Internet, or any server.
It goes with out saying, people tend to find it hard to explain (and the ones I've had to were from a technical background no less). This is particularly true when it comes to the command line tools, but GUIs fair no better. One very key problem is junk getting uploaded, I suggest if you're going to use this as a collaborative system you make sure to set some very clear rules, as seeing unnecessary junk files in the repositories is infuriating.