Here is a concrete example which may be useful to think about. It is not exactly the situation you describe but is similar. Let $k$ be a field of characteristic zero and let $R = k[x, y]/(y^2 - x^3 - x^2)$ be the ring of functions on a nodal cubic, and consider the maximal ideal $m = (x, y)$ of functions vanishing at the origin. Near the origin the nodal cubic crosses itself; loosely speaking, locally (in an analytic sense, e.g. when $k = \mathbb{C}$) it looks like two curves crossing each other. This crossing behavior is not directly visible in the quotient $R/m$, which is too local; it is also not directly visible in the localization $R_m$, which is "not local enough." By "directly visible" I mean the ring of functions doesn't resemble $k[x, y]/xy$, the ring of functions on two copies of the affine line crossing at the origin, and in particular it is still an integral domain so has no zero divisors. Geometrically the localization still remembers enough about the global behavior of the curve to remember that it is connected; it hasn't "zoomed in enough" to forget this.

The crossing is directly visible in the completion $\widehat{R}_m = \lim R/m^n$ (I'm not sure what the standard notation is for this). This completion is no longer an integral domain, since

$$x^3 + x^2 = x^2(1 + x) = \left( x \sum_{i \ge 0} { \frac{1}{2} \choose i} x^i \right)^2$$

is now a square, so we can now factor $y^2 - x^2(1 + x)$ as a difference of two squares, corresponding to the two branches of the curve $y = \pm x \sqrt{1 + x}$ which cross at the origin. So this "formal neighborhood" is smaller than the "Zariski neighborhood" and in this case it is now small enough that it resembles the "analytic neighborhood."

Here is a concrete example which may be useful to think about. It is not exactly the situation you describe but is similar. Let $k$ be a field of characteristic zero and let $R = k[x, y]/(y^2 - x^3 - x^2)$ be the ring of functions on a nodal cubic, and consider the maximal ideal $m = (x, y)$ of functions vanishing at the origin. Near the origin the nodal cubic crosses itself; loosely speaking, locally (in an analytic sense, e.g. when $k = \mathbb{C}$) it looks like two curves crossing each other. This crossing behavior is not directly visible in the quotient $R/m$, which is too local; it is also not directly visible in the localization $R_m$, which is "not local enough." By "directly visible" I mean the ring of functions doesn't resemble $k[x, y]/xy$, the ring of functions on two copies of the affine line crossing at the origin, and in particular it is still an integral domain so has no zero divisors. Geometrically the localization still remembers enough about the global behavior of the curve to remember that it is irreducible; it hasn't "zoomed in enough" to forget this.

The crossing is directly visible in the completion $\widehat{R}_m = \lim R/m^n$ (I'm not sure what the standard notation is for this). This completion is no longer an integral domain, since

$$x^3 + x^2 = x^2(1 + x) = \left( x \sum_{i \ge 0} { \frac{1}{2} \choose i} x^i \right)^2$$

is now a square, so we can now factor $y^2 - x^2(1 + x)$ as a difference of two squares, corresponding to the two branches of the curve $y = \pm x \sqrt{1 + x}$ which cross at the origin. So this "formal neighborhood" is "smaller" than the "Zariski neighborhood" and in this case it is now small enough that it resembles the "analytic neighborhood."