Circuits with large amounts of digital content are normally autorouted. Quite often power is also autorouted.
Analog circuits tend to follow pretty logical paths that lead to a specific flow. This makes them good candidates for placement by humans. Even these I will often hand place the components, and then autoroute that small analog section. If I have placed the components well it makes pretty good decisions on how to route the tracks, so I might only need to rip up a couple and reroute them, as I wanted to give a different priority than what the autorouter guessed.
For digital, differential pairs will want equal length routes. The better routing packages will recognize this, as you put some direction into the schematic to identify differential pair. Things like clocks will often have skew requirements, and a good package has you identify the clocks and their skew requirements, so the autorouter can optimize those.
Most high digital content will almost be fully autorouted. I come from an IC manufacturer, so half the die area on a chip was commonly digital. We would usually constrain where the inputs to the digital would come from, and where the outputs went to. Then we would define a placement area for all the gates (we called it the "sea of gates"), having given constraints to the autorouter, and hit route. 10's of thousands of gates would then route. It was not uncommon for an autoroute to start and run for 6-8 hours. But trying to route that by hand... would probably be measured in years for a person. It was common for a package to fail to autoroute a couple signals. This may then require shoving some of the analog around to open a little more room, and then shove some of the digital away to open a path for the remaining couple signals. At the end of a layout, I've seen it take a week to figure out how to connect 3 more signals.
So, autorouters are necessary for highly complex designs. I believe a license for the autorouter we were using for IC layout, designed to work with Cadence, was running about 50K a seat for a 1 year license.