From the question, we have:
$\begin{cases} 4-x\geqslant 0 \\ x-2\neq 0 \end{cases}$

Solving this gives us: $x\leqslant 4$ and $x\neq 2$,

Hence, the domain of the function is $\{x|x\leqslant 4,$ and $x\neq 2\}$,

Therefore, the answer is $\boxed{B}$.

By using the properties of square roots and exponents, we can find the domain of the function. This question tests the understanding of finding a function's domain and the meaning of square roots and exponents. It is a basic question.