Self-awareness is the experience of one's own character traits and skills.  It is not to be confused with consciousness in the sense of qualia.  While consciousness is being aware of one's environment and body and lifestyle, self-awareness is the recognition of that awareness.  In other words, self-awareness is how an individual consciously knows and understands their own character, feelings, motives and desires. 
Dr BB explains both self-reflection and rumination are examples of adaptive behaviours, so it is not a question of whether they are good or bad, but whether they are useful for us.  By adapting self-reflective behaviours, it most likely will help us cultivate or strengthen self-awareness.   
Self-reflection is associated with the ability to exercise more control over one’s feelings and behavior and generally to make better decisions in terms of how to respond. It is associated with the ability to learn and change. Conversely, rumination is a key contributor to mental health disorders such as depression and anxiety.