https://www.bloomberg.com/view/articles/2016-04-08/why-u-s-i...
The answer why is complex, but the two big factors seem to be red tape, specifically that standards are overengineered and it takes forever to get approvals, and lack of experience and existing construction infrastructure, so instead of having a well-greased machine like Singapore's that churns out line after line in rapid succession, you need to ramp up and ramp down for every single project.
The route would also be in a relatively recently built area, so it's not hard to imagine why it would be cheaper to plan and execute that a route under Manhattan.
I don't know very much about NYC, but the comparison does not seem fair.
I'm not a plumber, doctor, police officer, etc. Sometimes they are obviously right or wrong. Otherwise, I can try to second-guess them based on some limited or amateur knowledge here and there, but I know that when laypeople try that in regard to my profession, it very rarely helps them and can be a recipe for disaster. They don't even know what questions to ask.
In the end, IMHO we have to decide which experts to trust - this plumber or that one; this subterranean urban transport engineer or the other one. Thanks to the open societies in modern democracies, plus modern communication, we have a plethora of experts and voices to choose from, but it ain't perfect.
Which is the exact same way politicians can know whether projects are not mismanaged or over-budgeted by the construction companies, etc.
As far as I'm aware, the biggest issues with building deep tunnels like this is everything that gets in the way - other tunnels, utilities, old buildings that might fall down, and rivers. The Crossrail project in the UK has a high pice tag (£15bn) but is also much larger, and given how old London and its infrastructure is, I would imagine has had to cope with a similar number of obstacles, if not more. I imagine Paris is in a similar place given its age.
On what basis do you have any idea?
and that's precisely why oversight might fail and let corruption sneak in.
Many of the comparisons in the article seem to be similar to that, and with projects that seem far more complex, technically and politically, than what most of us deal with. Other than the fact that all the projects mentioned are called 'trains' or 'subways', and I assume are mostly underground, I don't have enough information to say they are comparable at all.
Sometimes, there is no way for the layperson to analyze the situation on their own.
EDIT: Minor edits