I found en excercise asking this question. I know that for proving the equivalence of NFAs and DFAs we can use the conversion through subsets, and that for proving the equivalence of nondeterministic TMs and deterministic ones we can build a 3-tape deterministic TM M which emulates the steps of a given nondeterministic one, let's call it N, proceeding this way:
- It copies the input string of N on the first tape
- On the 2nd tape, for each $i$th step of computation of N it produces a maximum of $d^i$ strings of length $i$ consisting of sequences of numbers between $i$ and $d$ representing the possible nondeterministic computations of N where $d$ is the nondeterminism degree of N
- For each of the above strings, it copies the content of the 1st tape on the 3rd tape and it tries every nondeterministic choice represened by its symbols, one by one
Basically M does a breadth first search on the tree of the computations of N, so if N has an accepting configuration on a computation path, surely M will find it because it will travel that path sooner or later.
So, for answering the question, I thought that the main reason is that a Turing Machines has a tape whereas a finite state automaton not so we can't use the same procedure used for converting a NFA into a DFA. Is it a sufficient answer? Should I mention that a Turing Machine isn't guaranteed to halt on any input too? How would you answer? Thanks in advance.