Is a Turing machine without the ability to write on blank cells less powerful than standard Turing?
The type of Turing machine you describe is a linear bounded automaton (it can only write on the parts of the tape containing the input). LBAs are the acceptors for context-sensitive languages so to find a specific example of a problem that can't be solved with this restriction but can be solved in general by a Turing machine, you just need a language that is decidable but not context-sensitive. 

The example given on Wikipedia is: 


  An example of recursive language that is not context-sensitive is any recursive language whose decision is an EXPSPACE-hard problem, say, the set of pairs of equivalent regular expressions with exponentiation.


For more examples, see Is there an example of a recursive language which is not context sensitive?