Buddhism vs Advaita Vedanta I understand that Advaita practitioners believe that we have an eternal self (consciousness) that is one with everything but that we are under an illusion that we are separate and through enlightenment we can realise the truth.
Theravadin Buddhists on the other hand believe that there is no 'self' but that we are under the illusion that there is one because we are too distracted to see that what we call self is nothing more than a bunch of impermanent processes i.e. the 5 skandhas. 
I have issues with both beliefs. 

With Advaita it doesnt matter how many Rupert Spira videos I watch I just dont see any evidence of anything living on beyond the death of my brain. Unless some penny drops and I finally get it somehow, my view won't change any time soon. Sure there is the sense of an observer when we sit and watch the thoughts etc but there is no proof that this is an 'eternal self' and not just some process of the brain being able think about thought.

With Buddhism I have begun to grasp the concept of Annatta because it kind of makes sense that all these different things are arising and passing however I have failed to hear any clear explanation about rebirth.  If there is no self then what is it that is reborn?  I know this question has been asked a zillion times but all I ever hear from Buddhists is that lame analogy about a candle flame going from one candle to the next which explains nothing.  If this illusion of self is not what gets reborn therefore I will not even remember my past life then why does it even matter at all? Also what do Buddhists believe about the observer of experience? They teach to sit and observe what is going on internally so what do they believe is observing? Nothing? Nothing is observing something?  The Advaita school of thought say that Buddhists are overlooking the truth. That the reason they think there is no self is because the self cannot look at itself just as eyes cannot look at themselves.
'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.