Buddhism vs Advaita Vedanta
'Re-birth' is merely the re-arising of the 'self-belief' or 'self-concept' (in this life). To quote:


  There is the case where an uninstructed, run-of-the-mill person — assumes form, feeling, perception, thought fabrication &amp;/or consciousness to be a 'self'. That assumption is a thought fabrication. Now
  what is the cause, what is the origination, what is the birth, what is
  the coming-into-existence of that thought fabrication? To an uninstructed,
  run-of-the-mill person, touched by that which is felt born of contact
  with ignorance, craving arises. That thought fabrication (of 'self') is born of that.
  SN 22.81
  
  
  
  The craving that makes for re-becoming — accompanied by passion &amp; delight, relishing now here &amp; now there — i.e., craving for sensual
  pleasure, craving-to-be, craving not-to-be: This, friend
  Visakha, is the origination of self-identification described by the
  Blessed One. MN 44


This 'self-belief' or 'concept' is a thought that is known or observed by 'consciousness'. Just as consciousness is not the thought of a 'dog' or 'tree', consciousness (which observes) is not the thought of 'self', 'I' or 'me'. The thought is a separate thing to the consciousness.

There is no evidence of anything living on beyond the ending of the activity of 'the' brain (not 'my' brain). The totally lame analogy about a candle flame going from one candle to the next which explains nothing was not an analogy spoken by the Buddha. Buddhism teaches the five aggregates are impermanent. When the activity of life (the aggregates) ends, that's it. If the mind can fully realise &amp; accept all life ends, it will be at peace. That is all that needs to be done. Please read the Anatta-Lakkhana Sutta and the Maggavagga and be free from doubt.