This is primarily a matter of definitions. Mainly, the issue is that, in standard mathematical notation, the question:
What are solutions to $2-x=-y$ where $y=\sqrt{x}$?
and the question:
What are solutions to $2-x=-y$ where $y^2=x$?
are different. The former uses the square root function (which has a branch cut) and the latter merely asserts that $y$ is some square root of $x$. The former implicitly takes that $y\geq 0$ from definition (or, $\frac{-\pi}2<\arg(y)\leq \frac{\pi}2$ if we're on the complex plane) whereas the latter makes no such assumption. So yes, in the standard definitions, there are solutions to the latter which are not solutions to the former.
In a question tagged algebra-precalculus, nothing of the below is likely to be relevant. What is more likely to be relevant is which of the two questions above are being asked (e.g. a word problem is almost never going to translate into the first one since the principal value of the square root doesn't really have physical interpretation, but a word problem might implicitly include another mechanism for choosing from the two solutions to the second one, such as "choose the smallest positive root"). In any case, let me demonstrate some of the use of considering other branches:
This all said, the main importance of branch points comes in when we consider the idea of "moving" a point around a function, rather than just plugging a single one in. That is, branch points are a type of problem that can occur to things that behave well locally, but poorly globally. For instance, suppose I give you some continuous function $\gamma(t):\mathbb R\rightarrow\mathbb C\setminus\{0\}$ and ask for some other continuous function $\sqrt{\gamma}$ which satisfies:
$$\sqrt{\gamma}(0)=\sqrt{\gamma(0)}$$
$$\sqrt{\gamma}(t)^2 = \gamma(t)$$
where the term $\sqrt{\gamma(0)}$ refers to the principal square root. This $\sqrt{\gamma}$ is uniquely determined, but it cannot be written as:
$$\sqrt{\gamma}(t)=\sqrt{\gamma(t)}$$
since, if $\gamma$ circles the origin - for instance $\gamma(t)=e^{i t}$ - then it crosses over a branch cut, and causes $\sqrt{\gamma(t)}$ to have a discontinuity, whereas $\sqrt{\gamma}(t)$ merely passes onto another branch of the square root. However, within some open interval of any $t$, we can choose a single branch of the square root function with satisfies our needs. We might view that this means we have a structure looks locally like a square root, but, globally, is passing between branches in a fluid way.
This issue arises more naturally when we do something like integrate $\frac{1}x$ on the complex plane, where we get $\log(x)$ as an antiderivative. Thus, a (contour) integral starting from point $a$ and ending at point $b$ should have value $\log(b)-\log(a)$ - but for this purpose, $\log$ is supposed to be a function whose derivative is $\frac{1}x$. It satisfies this, but has a branch cut. Thus, we necessarily have to consider the other branches when we use $\log$ for this purpose, since we might miss a multiple of $2\pi i$ if we naively use the principal value of $\log$ where we should have treated it as a multivalued function.
To use even more machinery, one might notice that, in complex analysis, branches are quite natural, since they would result from analytic continuation of $\log(z)$ or $\sqrt{z}$. The above two examples are more or less examples of that, where the continuation is implicit.
