Is there a difference between $\sin^2(x)$ and $\sin(x)^2$? Is there a difference between these? Or are they the same?
If they're the same, then why does $\sin^2(x)$ seem like its used more often?

The two possibilities are
$$
P_1 = \big(\sin x\big)^2
\\
P_2 = \sin\big(x^2\big)
$$
The convention (which confuses beginners) is
$$
\sin^2 x = P_1
$$
If you write
$$
\sin x^2
$$
or even
$$
\sin(x)^2
$$
we cannot tell which you mean.
