That's the second part of my course becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R, \mathfrak m, \kappa= R/\mathfrak m )$ be a local ring with max ideal $m$.
We can obtain two new rings $R^h$ (the Henselization) and $\widehat{R}_m$ the completion wrt $m$. Consider $R$ as a stalk of a nice enough scheme $S$ we can use these two constructions to obtain new new objects stalkwise: $S^h$ (here we have to differ between strict and "weak" Henselization) and the completion $\widehat{S}$. (recall $\widehat{S}$ is not more a scheme but a ringed space: localizations and completions not behave well to each other).
I would like to compare the main differences & (dis)advantages of completions & Henselizations from viewpoint of commutative algebra and (as well possible) geometric intuition.
The main motivation is that I often read comments like "in practice it's often nicer to work with Henselizations than with completions" in order to study the ring $R$ itself.
Question: Could anybody point out what are the advantages making Henselizations from certain viewpoint nicer to handle with then with completions?
In many comments the hand wavy arguments appearing in this context are like $\widehat{R}_m$ is much "bigger" that $R^h$ making it not "so easy handable like $R^h$". Could anybody bring more light in this formulation? When is mean by "bigger" (the added limits of Cauchy sequences I guess) but much more interesting what makes $R^h$ more "handable"?
The only point that I found out is that $Frac(R)=K \subset K^h$ stays algebraic and in many situations even finite. Is $R \to R^h$ also a finite $R$-module. In general that's not true for completions $ R \to \widehat{R}_m$.
Is this the only point making $R^h$ more handable than $ R \to \widehat{R}_m$?
What can we say about the geometrical part? The completion $\widehat{S}$ gives in certain way "analytic structure" to an (algebraic) scheme $S$ (very hand wavy; I know). About what kind of "geometry" one can think when one consider a henselization of a scheme (as for completion: local ring wise)? Some sources refer to "etale topology". It's a starting point of a huge machinery cumulating in stack theory.
Is there a geometric intuition how one can draw comparisons between endowings of $S$ "analytical structure" (as for completions) and with "etale topology" for $S^h$?
I know that there are a couple of questions here with similar titles (eg https://mathoverflow.net/questions/105381/henselization-and-completion , https://mathoverflow.net/questions/133499/completion-versus-henselization , https://mathoverflow.net/questions/217540/comparison-of-completion-and-henselization-in-class-field-theory ) but none of them deal with question of pure comparison of two constructions in the way I explained above.
Rmk: This is exactly the same question I asked some days ago in MO.
I can try to give a basic answer regarding the notion of "size". Henselization is a separable-algebraic extension, whereas completion need not be algebraic. For example, the completion of the rational function field $k(t)$ with respect to the $t$-adic valuation is the power series field $k((t))$. The transcendence degree of $k((t))/k(t)$ is uncountable.