actually, not true...

if you book your ticket on the train, it costs you more.

In my experience, booking in advance is advisable, simply because you can reserve seats (smoking, non-smoking, compartment, large open space, table, etc).

If you do not make your train because of the delay of your flight, but you are leaving on the same day/routing, most conductors will let you slide. You may lose your seat, but especially if you pull the tourist card, very few of them will actually make you pay the fare difference