What exactly differentiates Vipassana from Samatha meditation? What are the key aspects of Vipassana that are not present in Samatha? 

What is the main difference in the method from a meditator's perspective?

Labeling emotions and feelings are part of samatha or vipassana? Does Vipassana allows emotions to rise more freely to check it for what it is while Samatha tries to avoid paying attention to it by returning to the breath process?
What are the key aspects of Vipassana that are not present in Samatha?

samatha means tranquility - it is a necessary aspect of any wholesome meditative practice.

vipassana means seeing clearly or in a special way - it is a quality specific to Buddhist meditative practice.

Meditation for the purpose of seeing clearly requires one to focus on ultimate reality; the only way to understand reality is to observe it. Any meditation practice that does not take ultimate reality as an object is called "samatha meditation", because it leads only to tranquility, not insight.

Besides the difference in meditation object, meditation for insight will also obviously have different results; it will be less tranquil on the whole, as one is forced to experience all the inherent problems with ultimate reality, specifically that it is impermanent, unsatisfying, and uncontrollable.

What is the main difference in the method from a meditator's perspective?

There is no difference in the method, necessarily; the only difference is in the object.  As the Visuddhimagga says:


  But one whose vehicle is pure insight, or that same aforesaid one whose vehicle
  is serenity, discerns the four elements in brief or in detail in one of the various ways given in the chapter on the definition of the four elements (XI.27ff.).
  
  Vism XVIII.5 (Nyanamoli, trans)


Meaning the methodology is the same, but one's focus shifts to ultimate reality.

Labeling emotions and feelings are part of samatha or vipassana? 

Since emotions and feelings are a part of ultimate reality, this would be considered vipassana meditation. Many people say otherwise; I can't help but argue that they are wrong. The difference isn't the technique, it is the object.

Does Vipassana allows emotions to rise more freely to check it for what it is while Samatha tries to avoid paying attention to it by returning to the breath process?

samatha meditation has the potential to lead to avoidance, since it generally seeks out heightened states of concentration that are impossible when the focus is ultimate reality. The point is that one can only understand reality if one takes it as a focus; if you are unable to come to terms with reality, you will instead incline towards avoiding it because it is uncomfortable, preferring a single, stable, satisfying, controllable illusion to the harsh reality of the universe. This is a potential difference between the two types of meditation.

It is not that samatha meditation is bad or useless, just limited and posessing a potential danger, as the Buddha taught:


  “And what, bhikkhus, is the gratification in the case of feelings? Here, bhikkhus, quite secluded from sensual pleasures, secluded from unwholesome states, a bhikkhu enters upon and abides in the first jhāna, which is accompanied by applied and sustained thought, with rapture and pleasure born of seclusion. On such an occasion he does not choose for his own affliction, or for another’s affliction, or for the affliction of both. On that occasion he feels only feeling that is free from affliction. The highest gratification in the case of feelings is freedom from affliction, I say.
  
  [same with 2nd - 4th jhanas]
  
  “And what, bhikkhus, is the danger in the case of feelings? Feelings are impermanent, suffering, and subject to change. This is the danger in the case of feelings.
  
  -- MN 13 (Bodhi, Trans)