If, say, my laptop, with its 16GiB of RAM (meaning finite number of possible states), was actually Turing complete, that would have profound implications for mathematics. That would mean that there was an upper bound for the memory required for computation. But that's obviously not the case, because my laptop isn't Turing complete.
Does this affect your daily life as a programmer? Well, no. But does the distinction between "Turing machine" and "Turing-machine-like system but with finite memory" matter? In some contexts, yes. There are people whose profession it is to think about computation with more rigour than us programmers.
(FWIW, in most languages, while implementations aren't Turing complete, the language spec is. C is the odd one out by requiring implementations to impose an upper bound on the amount of memory which can be addressed.)