In this question, I saw that there was no difference between $\sin^2(x)$ and $\sin(x)^2$.
However, I am curious what's the difference between $2\sin(x)$ and $\sin(2x)$?
Notice that $2\sin x$ means take the value $\sin x$ and double it. Where as $\sin 2x$ means take the value $x$, double it, and then apply the $\sin $ function to that value.
Also, I do not believe it is universally accepted that $$\sin^2 (x) = \sin (x)^2,$$ since $\sin (x)^2$ can be interpreted as $$\sin ((x)^2).$$ For a less ambiguous case, take for example $\sin (2xy)^2$ is can easily be interpreted as $$\sin 4x^2y^2$$ or $$\sin^2 (2xy).$$ So I believe there is a difference.
For the sake of clarity, $\sin (x)^2$ should really be written as $$(\sin x)^2$$ if you mean to convey $\sin^2 x$.
$2\sin x$ means first you take the sine of $x$. Then you multiply the result by 2.
$\sin 2x$ means first you multiply $x$ by 2. Then you take the sine of the result.
$\sin 2x$ can, and sometimes should, also be written as $\sin(2x)$.
It is not ok to pull constant multiples out of trig functions.
Really, $\sin(x) ^2 = \sin^2(x)$ only because we adopted the function notation used prior, namely $f(x) ^2 = f^2(x)$. In most cases I still use the former to avoid confusing differentiation or iteration, but trig functions still use the latter. Now, this notation only holds for powers... When you say $2\sin(x)$ you are making a sine wave with an amplitude of $2$. When you say $\sin(2x)$ you are talking about a sine wave with twice the frequency of the standard sine wave.