Consider the function, $$f(x)=x^{x^{x^{x^{.^{.^{.{^.}}}}}}}$$While differentiating it with respect to $x$ (provided that $f(x)$ evaluates to a finite value), we were made to write $f(x)=x^{f(x)}$.
My question is: Why we can't write it as $f(x)=(f(x))^x$ and then differentiate implicitly? Both the ways of writing mean the same thing.
You have to be a bit careful here. The point is that, for each $x$, $f(x)$ is defined as $f(x) = \lim_{n\to \infty} p_n(x)$, where $p_n$ means applying $n$ times "$x$ to the power", e.g, $p_2(x) = x^{x^x}$. (Note that is is not the same as $(x^x)^x$!) Then it follows that
$$ f(x) = \lim_{n\to \infty} p_n(x) = \lim_{n\to \infty} p_{n+1}(x) = \lim_{n\to \infty} x^{p_{n}(x)} = x^{\lim_{n\to \infty} p_{n}(x)} = x^{f(x)}. $$
We cannot do a similar sequence of equalities to arrive at your expression $f(x) = f(x)^x$.