However, in physical reality, there exist no Turing machines. There are only finite state machines whose behavior emulates a quasi-Turing machine with a finite tape (or an analogous crippled finite version of some other Turing-complete theoretical construct).

Now, every finite-state machine can be implemented using a lookup table and a transition function that simply performs a lookup based on the current state and input. Any computers we have now or in the future can be only clever optimizations of this model. For example, von Neumann machines (i.e. computers as we know them) avoid the impossibly large lookup table by implementing the transition function in the form of a processor that, at each step, examines one small subset of the state and produces a new state that differs only by another small subset based on simple rules. (I'm describing the effect of a single machine instruction, of course.)

So, the question is: what exactly makes a lookup table deficient compared to a "real" computer, whatever that might be?