Why are Linearly Bounded Turing Machines more powerful than Finite State Automata?
The linear bounded Turing machine is restricted to a tape whose length is a linear function of the length of the input.

If the length limit were a constant, then the machine would be no more powerful than a DFA. However, a DFA cannot grow more states to cope with a longer input, which in effect the LBTM can do (taking the state to be the entire machine configuration.) So the LBTM is strictly more powerful.