Having some version control system helps in any, many cases:
Single developer, single branch
The most basic task that each version control system has to perform perfectly if it wants to call itself version control is to be able to go back to specified version of a project. If you made mess of things, you can got to previous version. You can examine some previous version to check how it was done then (for example how it was before refactoring, or before removing some code/file).
Version control systems take much less disk space compared to simply saving backup copies with specified date, because they use deltaification (storing only differences from previous version) and compression. Usually backup systems are means to store last N versions of a project, sometimes with N=1 (only previous version) while version control systems (VCS) store all the history of a project. Knowing Murphy a while after deleting Nth last version you would realize that was the version you want to examine.
Additionally going back to some last version is easy and automated. You can also examine how single file looked like at some past version, and you can get differences (in diff format) between current state and some past version. You can also tag (or 'label') versions, so you can refer to past version not only by date, or by being nth version from current one, but also by symbolic name, for example v1.2 or v1.2-rc0.
With version control system you can examine history to remind you why (and how) some piece of code (some part of a given file) arrived at current state. Most VCS allow to examine line-wise history of a file, i.e. annotating each line of a file when it was changed, in what commit, and by whom (the command is named annotate, blame or praise depending on VCS). In some VCS you can search history for a version (revision) which introduced given fragment of code (e.g. called 'pickaxe search' in Git, one of VCS).
For this feature to be really useful you have to maintain some discipline: you should describe each new version (each new revision / each new commit) writing down why the change was made. Such description (commit message) is very useful, but it doesn't have natural place in backup system.
This feature of course is even more useful if you are not the only developer...
Using version control system allows for alternate way to find bugs in the code, namely by searching history to find version which introduced bug: bisectiong history. When you find revision which introduced bug, you would have limited (in best case: very limited) area to search for bug, because bug has to be in the difference betwen last working version and first version with a bug. Also you would have description of a change (a commit message) to remind you what you wanted to do. This feature is also called sometimes diff debugging. Modern version control systems (VCS) have support for automated (or semi-automated) searching the history by bisecting it (dividing history in half, finding which part contains bug, repeat until single responsible version is found), in the form of bisect (or similar) command.
For this feature to be really useful you have to maintain some discipline: you should commit (save changes / put given state in version control system to remember) single change, dealing with only one feature, with only small difference from the previous version; i.e. commit often.
Most version control systems offer various hooks which allow for example for automated testing, or automated building of a product... or simply reminding you that you do not follow coding standard (coding guidelines).
Single developer, multiple branches
Version control systems allow to create multiple alternate parallel lines of development, called branches (or streams, or views). Common case is having development branches, i.e. having separate branch for unstable development (to test new features), separate branch for stable (main, trunk) version which is (or should be) current working version, and one on more separate maintenance (fixup) branches.
Having maintenance branches allow you to do bugfixes and generate service packs / minor version with corrections to some released version, without need to worry about interference from the new development. Later you can merge maintenace branch into stable, or pick bigfix from maintenance branch into stable and development branches (if further/other development didn't fix bug independently).
Modern VCS (here modern means that both branching and merging branches is easy) allow to go a bit further, i.e. generate separate branch for working on a separate feature (so called topic branches). This allow you to switch between working one one feature to working on other feature (and not only switch from eveloping new feature to working on urgent requested bugfix).
If you are developing your product based on source of some other (usually third party) product, you really should use vendor branches to be able to easy integrate new version of a product from vendor with the changes you made. Admittedly this is no longer purely "single developer" case.
Multiple developers
Using version control systems brings even further advantages if there are more than one developer working on the same project. VCS allow for concurent (parallel) development without worrying that somebody would overwrite your changes, or does not take your changes into account. Of course using version control system is no substitute for communication.
All of the above features are even more important in the multiple-developer case: examining who generated given change, who last changed the code (aka. who broke the build), finding a bug in code not written only by you.
