An isometry is a way of moving the manifold rigidly within itself, and is inherently an active transformation (a diffeomorphism).
From <PERSON>'s Lecture Notes on General Relativity:
diffeomorphisms are just active coordinate transformations... We say that a diffeomorphism $\phi$ is a symmetry of some tensor $T$ if the tensor is invariant after being pulled back under $\phi$... The most important symmetries are those of the metric... A diffeomorphism of this type is called an isometry.
In the formula $$g^\prime_{ij}(x')=g_{ij}(x'),$$ it is important to note that the argument $x'$ is the same on both sides -- i.e., the original and transformed metrics are evaluated at the same coordinate values and hence at different physical locations. Of course, there is technically always the trivial isometry $x'(x) = x$. 578
But in general, each point $P$ is mapped by the isometry to a different point $Q$.
(A passive version with $g^\prime_{ij}(x')=g_{ij}(x)$ -- note no prime on right-hand side -- would be different and less interesting. This says that the metric looks the same in the old and new coordinates at the same point $P$. Such nontrivial coordinate transformations exist for any manifold -- e.g., adding a constant to each coordinate. Nontrivial isometries, on the other hand, exist only for special manifolds.)
We can describe an isometry without even using coordinates. Given any points, curves, and other shapes of interest in a manifold, all their metrical properties and relations (such as lengths and angles) are unchanged when they are transformed into new points, curves, and other shapes by the isometry.
We can see that an isometry is very special, since a general manifold with metric will have lots of irregular (but smooth) "bumps and wrinkles" as measured by curvature, such that any point can be uniquely identified by the pattern of bumps and wrinkles around it. There will be no way to transform the manifold nontrivially while keeping these patterns the same -- i.e., no possible rigid motion of the manifold within itself.
A continuous family of isometries is particularly interesting, and leads to the notion of a Killing field. 578