I know that Maclaurin series are a special case of Taylor series where we set $a=0$, and it is useful for exponential functions (for example). But my question is: Are Taylor series (evaluate at some other point different to zero) really useful in practice?
An important reason is that the function might not be defined at $0$. Consider the $log$ function.
See this previous questions: taylor series of ln(1+x)?.
As mentioned in the comments, you might need to approximate the function in a region not near $0$. The Maclaurin series might converge very slowly or not at all. You could look for nearer points st which it was easy to calculate the Taylor coefficients.