Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?
To complete the other answers: I think that Turing Machine are a better abstraction of what computers do than finite automata.
Indeed, the main difference between the two models is that with finite automata, we expect to treat data that is bigger than the state space, and Turing Machine are a model for the other way around (state space >> data) by making the state space infinite. This infinity can be perceived as an abstraction of "very big in front of the size of the data". When writing a computer program, you try to save space for efficiency, but you generally assume that you won't be limited by the total amount of space on the computer. That is part of the reason why Turing Machines are a better abstraction of computers than finite automata.