I don't disagree that a train is better in many situations. I'm more curious about situations where a city is prepared to make a 1B+ capital expenditure. For example boring a tunnel for a train vs. a dedicated bus lane on the highway + exit.

One train can have a capacity of 1000 people. If you run these at 3 minutes interval, you have a throughput of 20,000 people per hour.

One bus takes less than 100 people, and you cannot run 200 buses per hour. Actually about 20 buses per hour is pretty much at the upper limits.

These are rough numbers, but if you need to move more than 2000 people per hour, trains can have the capacity but buses don't.

Is that dedicated lane cheaper? In many cities that are facing a traffic crunch, they've already made all of the "easy" road improvements -- for example, 101 in the SF Bay Area is pretty much all built in to the center median, so adding new lanes means adding outside lanes (which requires rebuilding every bridge and intersection).

It'd be cheaper, of course, to take an existing lane and convert it to a bus-only lane, but that's politically infeasible.