Two "different" definitions of $\sqrt{2}$
It helps to view each construction in the context of ordered (or orderable) fields.
The algebraic definition describes a field, but $F=\mathbb{Q}[x]/(x^2-2)$ is a bit more than just a field: it's an orderable field. There are exactly two ways to make $F$ into an ordered field, determined by which square root of $2$ we pick to be positive.
On the "geometric" side, an ordering is exactly the additional data provided by Cauchy sequences (or Dedekind cuts, or etc.)! There is a certain set $X$ of equivalence classes of Cauchy sequences such that when we look at "$\mathbb{Q}+X$" and forget the ordering, we get an algebraic structure isomorphic to $F$. So basically, when we add $\sqrt{2}$ to $\mathbb{Q}$ in "the geometric setting" - and add a few more points to get good closure properties - we wind up with strictly more information than is provided by the purely algebraic construction of $F$. Moreover, the "extra points" we need (basically $X\setminus\{\sqrt{2}\}$) are determined in a straightforward way, so it's not that much extra information.
The relevant theorem here is: "For any set $Y$ of equivalence classes of Cauchy sequences, there is a smallest set $X_Y$ of equivalence classes of Cauchy sequences such that $Y\subseteq X_Y$ and "$\mathbb{Q}+X_Y$" is a field, and moreover $X_Y$ has a snappy description" (that last bit being a good exercise). Above, $X=X_{\{q\in\mathbb{Q}: q^2<2\}}$.
In this sense, the "geometric" approach provides strictly more information than the algebraic approach. On the other hand, it's not much more information: the two orderings on $F$ are isomorphic in the obvious way, so up to automorphism $F$ is a uniquely orderable field. So ultimately the two approaches aren't that far apart.
Incidentally, it's worth noting just for fun that $\mathbb{R}$ is in fact a truly-uniquely-orderable field since we can recover the ordering from the algebraic structure: $a\le b$ iff $\exists c(c^2+a=b)$. This is not in general true, to put it mildly, but it's cool.
The fundamental difference between them is that they generalize differently: Cauchy sequences generalize to arbitrary metric spaces without any required algebraic structure, while ring extensions/quotients generalize to arbitrary rings without any required geometric structure.