Difference between read-only Turing machine and non-erasing Turing machine
A non-erasing TM has a tape alphabet $\{0,1\}$ where $0$ acts as blank. The $0$ can be rewritten into $1$, but that symbol cannot be "erased", i.e., written back into $0$.

This restriction seems to be equivalent to symbols that can be rewritten only a finite number of times. TM's do have the capability of interpreting a finite segment of tape as one symbol. Then $0000$ can be rewritten into $0001$ then into $0011$, into $0111$, into $1111$. A little modulo counting interprets these segments as $0,1,0,1,0$.

Recall that a TM is equivalent to a finite state machine that has a queue as basic datastructure. It can read and delete on one side, and write on the other. To see that this is equivalent to a TM a rotation trick is done: the queue holds the tape in a circular fashion, but can only move on the tape in one direction. To move into the other direction just move all the way.

This rotation trick can also be done on the tape of a TM, and needs only finite rewrites on the same position. One needs to be able to rewrite the tape from $Ax$ into say $xB$ where $A,B$ are symbols and $x$ is a string.