How does *Buddhist* meditation differ from others and lead to awakening? The practice  of meditation is central to certain Buddhist traditions, e.g. Vajrayana, Dzogchen, Zen, important for recognizing Buddha nature.

Furthermore, Vajrayana and Theravada traditions assert that the meditation serves as a basis for realizing selflessness:


  the purpose of meditative stabilization is to serve as a basis for
  achieving supramundane special insight realizing selflessness, the
  emptiness of inherent existence, through which afflictive emotions can
  be removed completely and forever
  
  -- Dalai Lama
  
  So what is the purpose of satipaṭṭhāna? The purpose is to see
  anattā, that there is no self, no me, nor anything that belongs to a
  self.
  
  -- Ajahn Brahm


But dhyana and samadhi are not unique to Buddhism, also found in schools such as Hinduism which do not lead to realization of no self. 

And even highly accomplished Buddhist meditators have found that with a radiant mind in meditation, the sense of self can still fail to be uprooted. 


  At that stage the mind was so radiant that I came to marvel at its
  radiance... 
  
  “If there is a point or a center of the knower anywhere, that is an agent of birth.”
  
  -- Ajahn Maha Boowa 


How exactly does Buddhist meditation practice lead to realizing selflessness, versus other forms other forms of meditation which do not? What exactly makes Buddhist meditation different?

What made the Buddha's meditation technique differ from others' (e.g. Alara Kalama and Uddaka Ramaputta) which did not lead to enlightenment?  (Please be more specific than "Middle Way").
Very good question.

Buddhist meditation is based on the understanding of the mechanism of arising and cessation of suffering. Suffering arises whenever there is a conflict between "is" and "should". Craving for any "should" because of attachment to that (usually to a concept within a framework) is the generator of suffering. Letting go of the "should", or even better, transcending the boundaries of a framework that served as the foundation for the concept, leads to cessation of suffering.

Buddhist meditation involves methodically applying the above principle from coarse to progressively finer "shoulds".

As you thus let go of these attachments, preconceptions, prejudices, and overgeneralizations - the notion of self dissolves as the hidden counterpart of these overgeneralizations.

However, I think you got the relationship wrong way. It's not that the peace of no-attachment (no-should, no-overgeneralization) is a step that leads to selflessness which then leads to Enlightenment. It is the other way: freedom from attachments, preconceptions, prejudices, and overgeneralizations -- including the overgeneralization known as "Self" - leads to Enlightenment and Peace.

Non-Buddhist meditation is either trial and error, or something trivial like gazing at the candle or running water, or it is brute-force stopping the thoughts, or it is visualization of some images that has useful side effects, or working directly with psychosomatic energies. The last two types were later repurposed by Tibetan Buddhists and integrated into Buddhist framework of progressive attainment of suchness.