Difference between linear bound automata and a Turing machine
Notice that linear bounded automata are precisely all TM's who use $O(n)$ space.
Now, by the space-hierarchy theorem, for any $f$ where $n=o(f)$ (for an extreme example, $f(n)=2^n$) we would have $DSPACE(O(n))\subsetneq DSPACE(O(f))$.
Thus there are languages who require space complexity bigger than $O(n)$, and therefore are not solveable by linear bounded automata but can be solved by a ($O(f)$ space) turing machine.