If you run an entire website off of a single machine, and that machine is more than enough to handle your site, then kudos, images are not a bottleneck that needs solving right now. Forget about S3 for now.
However, as your server gets busier, you will want your server to be spending all of its time doing server things. Transferring static content like flat HTML files and images is an easy, dumb job, and wasting precious active connections, bandwidth, and CPU cycles on them is no good. By switching to S3, your server can concentrate on doing what's important, which is whatever your program actually DOES.
S3 also has benefits of being distributed around and attached to what's probably a fatter pipe than your server, which means the images will show up slightly more quickly on your client's machines, so that's an added bonus.
S3 is also backed up, which means that it makes for a pretty nice place to store pretty much any private data under the sun, in addition to stuff that you want to serve to others (although don't confuse the permissions settings between those two things -- in fact, you may want to use separate accounts entirely).
S3 is also nigh-infinite, which means that if you want to let users upload files to your site (profile images, attachments, etc), S3 is a great choice so that you don't have to constantly worry if your server is going to run out of disk space (obligatory $$$ warning here).
But like I said at the top, if you're a one-server setup with a handful of users, none of this really matters. It's a tool like any other, and it may not be something you need yet.