I cannot quite feel the difference between two models.
The first one is read-only Turing machine which is basically the same as Turing machine with write protected input. See this question for example: Single-tape Turing Machines with write-protected input recognize only Regular Languages
It was said that it indeed recognizes only regular languages because there is no way to copy an input without rewriting it somewhere.
And there are non-erasing Turing machines that (in binary representation) cannot overwrite $1$ with $0$. This is equivalent to full power Turing machine as it was explained here Non-erasing Turing machines and loss of generality and in the paper http://arxiv.org/pdf/1304.0053
But what is the difference? We still cannot copy some strings like $1111$ due to lack of overwriting power.
So why it is weak when we cannot write only on input but strong when we cannot overwrite $1$ with $0$ everywhere? Why we can encode strings in one case but not the other?