Is a computer equivalent to a Turing machine or linear bounded automata?
A machine bounded by $\Theta(1)$ memory is a finite state automaton (As a matter of fact, proving that $\textsf{DSPACE}(\Theta(1)) = \textsf{REG}$ is a classical exercise).

We say that "Turing machines can do whatever computers can" because, as long as we keep in mind that real computers are subject to limitations in computational power, it's more useful to think of computers as imperfect implementations of a universal Turing Machine rather than perfect implementations of a finite state automaton.