Since you mentioned saturation in your question, perhaps it may be helpful to explain one basic difference between the reals and the hyperreals in these terms. Note that the real number field does not satisfy saturation: you can always find a real number $x$ satisfying finitely many relations of type $x>n$ for natural $n$, but not all of them. In the hyperreals, you can find a number $x$ satisfying all of the countably many relations $x>n$ for natural $n$. In fact you can find an $x$ satisfying countably many such relations even for hypernatural $n$. However, one cannot find an $x$ satisfying all such relations for hypernatural $n$ because there are uncountably many of them. In sum, the simplest model of the hyperreals (which is unique up to isomorphism if you assume CH) is countably saturated.
To respond to the additional questions posed by the OP:
(1) How does a non-standard model help us understand "ordinary" real analysis?
A lot has been written about this. I will respond briefly by citing one of my favorite examples, namely (Cauchy's) definition of continuity: $y=f(x)$ is continuous if every infinitesimal $x$-increment necessarily leads to an infinitesimal change in $y$. Many students find this more accessible than the epsilontic definition: $y=f(x)$ is continuous if for every $\epsilon>0$ there exists a $\delta>0$ such that for all $x'$, whenever $|x'-x|<\delta$ one has also $|f(x')-f(x)|<\epsilon$. There is a lot of educational literature on the students' difficulties with alternation of quantifiers.
The OP asked for a more advanced illustration. The notion of uniform convergence is one case where working in an infinitesimal-enriched continuum helps clarify the mathematics. In an A-continuum, the definition of uniform continuity in an interval $I$ is that $$(\forall\epsilon>0)(\exists\delta>0)(|x-y|<\delta\implies|f(x)-f(y)|<\epsilon).$$
This definition involves two variables ranging through the domain $I$ and in this sense is not "pointwise" (or "local") in the sense that ordinary continuity is a pointwise notion.
Meanwhile, an equivalent definition of uniform continuity can be given in a B-continuum which is local, in the following sense. Here we work with the natural extension $f^*$ of a function $f$. Given a point $c\in I^*$ of the extended domain $I^*$, we denote by $\mu(c)\subset I^*$ the set of points infinitely close to $c$. we say that $f^*$ is microcontinuous at a point $c$ if $f(\mu(c))\subset \mu(f(c))$. Informally, if $x\in I^*$ is infinitely close to $c$ then $f(x)$ is infinitely close to $f(c)$. (This was essentially Cauchy's definition of continuity.)
Now it turns out that one give the following local definition of uniform continuity of $f$ on $I$. Namely, $f$ is uniformly continuous on $I$ iff for every $c\in I^*$, the function $f^*$ is microcontinuous at $c$. Then showing the failure of uniform continuity of $y=x^2$ on $\mathbb{R}$ becomes a trivial exercise of checking that the function fails to be microcontinuous at a single infinite point.
(2) How do we use a tool like model theory to clarify and better express ambiguities in "simple" concepts like considerations about completeness in Archimedean as against non-Archimedean fields?
The Archimedean property can be written as follows:
$$(\forall\epsilon>0)(\exists n) (n\epsilon>0)
$$
If one interprets $n$ as being a natural number, then this property fails for the hyperreals (choose $\epsilon$ to be infinitesimal). If one interprets $n$ to be an arbitrary hypernatural, then this property is satisfied by the hyperreals.
One can also formulate this in terms of quantification over sets. Then the property will fail when quantification is over arbitary sets, but will be satisfied when quantification is over internal sets.
I could write more but I will stop at this point to see if this is the direction the OP had in mind.
