Well on smaller designs it's a no-brainer. You, as a designer, can see all the things an autorouter can't -- how your analog signal flow needs to lay out (using short, compact routes), your power layout (critical even in a small design) and a lot of other things like maximizing signal traces and minimizing vias.
Autorouters do lots of other bad things but those are the major ones. Personally I'd never use an autorouter unless the design was several thousand nets and maybe 6-8 layers -- and with a design like that you'd really need an "industrial strength" autorouter like Specctra. For anything I'm putting on a 2 or 4 layer board I'd hand route it.
I know a lot of other people will probably disagree with me but I've never seen a board that meets my "hand route" criteria yet that looked better from an autorouter vs. hand routed.
Here's an example board
. It's a 4-layer board, about 800 nets and a mixture of analog, digital and power circuits (like a lot of "hobby" projects, only bigger). I doubt many autorouters could have completed this board, let alone done a better job than hand routing.
Here's another example board
that's much closer to what a hobbyist might do. It's a 2 layer board and is about 2.75" x 1.25". It also has analog, digital and power components.
These are pretty typical of the board designs I do. I don't believe any autorouter would do a better job or save much (if any) time in the PCB design. They also couldn't handle the mix of analog, digitial and power very well.
I think a lot of people figure that if they have an autorouter they must use it and that it will automatically save them time and produce a better result. This is my perception of the online discussions I see but not the results (in my experience). A lot of people should really consider hand routing their boards unless there's an overriding reason not to. Ask any experienced professional PCB designer and he'll tell you that the hand routed boards are generally a lot better layout.
I'm not "anti-Autorouter", I just think there's a time and place for them and that the vast majority of boards are better off routed by hand than autorouted.