How does *Buddhist* meditation differ from others and lead to awakening?
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.