the difference between Positive Thinking and Positive Believing?

Believing refers some principle and thinking will help to prosper.

Positive thinking belongs to our brain whereas the positive believing belongs to heart.

The term 'positive' has been used both with 'Thinking' and'believeing'.  The term 'positive'  implies an optimistic and constructive attitude.  'Thinking' is mental process for forming opinion and taking decisions based on inputs received by brain.  'Believeing'  is the act of agreeing to others' opinions.  The belief may be blind or based on self reasoning. Taking in view the definitions of 'positive', 'thinking' and 'believeing' , the required conclusions may be arrived at.

Positve thinking never stops a person from thinking anything good in this life. He will forecast the good aspects of life.
Positive belief will make a person belive anything in good sense or in positive sense. He will never except the harsh facts of life.

There is no difference at all. Both amount to Optimism

Thinking imbibes belief. So, I guess, more or less, both should mean the same. If one things positively, there is a belief that something constructive or nice would happen. A positive start to any thing puts in the confidence of believing.

Positive thinking may be that in a negative situation you are thinking or hoping that something positive may happen, positive believing is that you believe that something positive will happen and you are not hoping but you strongly believe that something positive will happen.