I would recommend the train for the following reasons:

A) Aman recommends it (hugely important)
B) Aman takes "possession" of you while you are still in Delhi. You get to meet your batman and spend as little or as much time chatting with him as you'd like. He will pamper you on the entire journey with snacks, lunch, chai, etc.
C) The train has sleeper seats so you can get fully flat and catch up from your jetlag.
D) The train is a cultural experience and the main source of transportation in India
E) There is lots of interesting local scenery outside the windows
F) You don't have to risk late flights or missed connections