Church–Turing thesis and infinite Turing machines
The Church–Turing thesis is the following philosophical claim:

Our intuitive notion of computability coincides with the mathematical notion of computability.

Here the mathematical notion of computability states that a language is computable if some Turing machine computes it, and the intuitive notion of computability is necessarily vague.
Turing machines with infinitely many states can compute any language (exercise), and so are more powerful than the usual Turing machines, which can only compute computable languages. We know that not all languages are computable. Indeed, almost all languages are uncomputable, since there are only countably many computable languages, but uncountably many languages. Also, certain concrete languages such as the Halting Problem are known to be uncomputable.