What are the main arguments in favor of the Anātman view over the Ātman view? The main difference (I know) between Buddhism and Hinduism is that the first defends Anātman theory whereas the second defends Ātman theory.    

What are the main arguments in favor of the Anātman view over the Ātman view?
The idea of Atman tends to be defined as having three characteristics:


permanent
controllable
satisfactory


When it is any of those things, it can be seen as Atman.

In SN 22.59 the Buddha explains that neither form, feeling, determination and consciousness have those characteristics:


  Bhikkhus, form is not-self. Were form self, then this form would not lead to affliction, and one could have it of form: 'Let my form be thus, let my form be not thus.' And since form is not-self, so it leads to affliction, and none can have it of form: 'Let my form be thus, let my form be not thus."
  
  "Bhikkhus, how do you conceive it: is form permanent or impermanent?"
  — "Impermanent, venerable Sir." — "Now is what is impermanent painful
  or pleasant?" — "Painful, venerable Sir." — "Now is what is
  impermanent, what is painful since subject to change, fit to be
  regarded thus: 'This is mine, this is I, this is my self'"? — "No,
  venerable sir."'


note: I've translated it as: "Unsatisfactory vs satisfactory"; the quoted text translated it as "Painful vs Pleasant" and I've also seen "stress-full vs stress-less." I hope the point is clear.

It's also something that becomes apparent after meditating and following the path, when the Buddha asks Annanda, what the 10 perceptions gained by mindfulness of in - and out-breathing are, he answers pertaining to non-self:


  "And what is the perception of not-self? There is the case where a monk — having gone to the wilderness, to the shade of a tree, or to an empty building — reflects thus: 'The eye is not-self, forms are not-self; the ear is not-self, sounds are not-self; the nose is not-self, aromas are not-self; the tongue is not-self, flavors are not-self; the body is not-self, tactile sensations are not-self; the intellect is not-self, ideas are not-self.' Thus he remains focused on not-selfness with regard to the six inner &amp; outer sense media. This is called the perception of not-self.


On a more secular note, Sam Harris, who has been inspired by Buddhism and tried meditation, talks about this realisation as well:


  Occasionally, a breakthrough did occur: In a moment of seeing, for instance, there would be pure seeing, and consciousness would appear momentarily free of any feeling to which the notion of a “self” could be attached. But then the experience would fade, and I couldn’t get back there at will. There was nothing to do but return to meditating dualistically on contents of consciousness, with self-transcendence as a distant goal.


I believe that those are the main argument in favour of the Anatman view:

1) understanding that experience is changing , uncontrollable and unsatisfactory

2) Seeing in meditation that when one is mindful, no sense of self arises.  

thus concluding that there is nothing that one can say about: 

"This is me, This is mine, I am this."