Here's a comparison. For the autorouted board, I had to route ground net manually for some reason (probably because I put in a ground polygon before feeding it to the autorouter), but that's mostly just via to ground plane. Otherwise, the router routed the whole board.
Well, I think your placement of parts is a bit wonky to start with. I consider myself quite a novice, but I tend to group like parts together. An example is C11 which I guess is a filter cap for a voltage divider of some sort ... maybe a temperature sensor? Anyway, I'd want to put it near the resistors its associated with and pretty that up first. By orienting them right, you can get some short traces to bind them all together and then have a couple traces that look like the inputs and outputs they are in the schematic.
On the other hand, the autorouter decided to run that trace right between the pins on the clock crystal capacitors. That looks like a recipe for trouble, with a 20MHz square wave capacitively-coupled so close to what seems to be a signal trace.
Consider also how you handled the high-current traces. I think the VCC routing under U6, U7, and U8 looks pretty good, but the wonky line around the outside of the connector doesn't. My own technique is to make copper pours of each power supply input (ground, ±V or just +V as necessary), often splitting them under a row of chips so the power pins are on the right side pour.
Overall, to put it rather bluntly, your hand-routing technique isn't very good, so rather than saying the autorouter beads hand routing, I'd say your hand routing technique is worse than this autorouter. There are boards that have some really elegant routing solutions that an autorouter would just not bother to do.
In the end, for small-signal, low-frequency boards, the prettiness of the underlying routing is not too relevant as you're not going to have issues with crosstalk and the like. As such, any old autorouter, as long as it connects all the nets, will do fine. But once you add a few high-current, high-impedance, or high-frequency traces, knowing how to route things elegantly and getting good at the TETRIS game of component placement will help.
As I say, I'm no wizard at this, but here's an example of a controller board I made for an electric conversion of a riding lawnmower. Just the top layer from OSHPark's render ... the point is the pours provide lots of connections at low impedance, and the traces are clear and straightforward. The components are laid out so functions are grouped. (The odd connectors at the bottom allow either screw-terminals or Anderson PowerPole connectors to be installed in the board.)