Yup, It does behave like a finite state machine.
if for (every permutation in tape string, every current head position, every next move) you create a unique state, you are besically creating a DFA like mechanism, as (next move/tape write) is determined by those variables and you have a state for every possible combination. And as the tape is finite, the no of state will also be finite.
the thing is-
- limited memory= Finite automata
- one time input scan + unlimited memory= push down automata
- multiple tape scan + memory which is linearly proportional to input size= Linear bounded automata
- multiple tape scan +unlimited memory= Turing machine
so, you might think that limited memory turing maching behaves like a Linear bounded automata but as the memory is limited and not growing with respect to input, it will not be as powerful as LBA but just a Finite automata.
see, if the input is given in the tape itself . so, with larger input, the tape length has to increase. which means you can scan the input multiple time+ you have space (memory) linearly dependent to the input length. so, the whole setup will be as powerful as LBA. If the input is given some other way and not in the "fixed sized" tape, it will be a DFA only.
But by the "finite tape" in the qustion, I assume you have some some fixed no of cell (K) irrespect to the input length. so for an input larger then K simbols, you have to provide some other way as the tape can hold only K simbols. in this scenario, the TM will not be more powerful than a DFA even if you can write the tape (as number of elements in $\Sigma$ is finite so only finite tape configuration is possible).