When Turing Machine behaves like Finite state automaton
Yes, this is (kind of) correct, in the following sense:

The set of languages that can be recognized by Turing Machines with a fixed tape is equivalent to the set of regular languages (i.e. the set of languages recognizable by finite automata).

The reason behind this is that such Turing Machines have a finite number of configurations, which is independent from the input. 

Note that crucially, for this to hold we need one tape for the input which is read-only, and a fixed-space working tape.

A bit more precisely, for such a TM, we can construct a 2-way DFA, by having a state for every possible configuration of the TM, as well as a 2-way read only input tape. The latter model is equivalent to DFAs, by a clever construction using "crossing sequences".

An important observation, however, is that these machines can be far more succinct than their equivalent DFAs (or even NFAs).