The following result is known as the Komlos-Major-Tusnady approximation:
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $(X_n)_{n \in \mathbb{N}}$ be an iid sequence of random variables on this probability space and let $S_n = \sum_{j=1}^n X_j$ denotes its partial sum process and let $p > 2$. If $\mathbb{E}(X_1)=0$, $\mathbb{E}(X_1^2)=1$ and $\mathbb{E}(\vert X_1 \vert^p)< \infty$, then there exists a probability space $(\Omega', \mathcal{A}', \mathbb{P}')$, a sequence $(S_n')_{n \in \mathbb{N}}$ on this probability space such that $(S_n)_{n \in \mathbb{N}} \overset{\mathcal{D}}{=} (S_n')_{n \in \mathbb{N}}$ and a Brownian motion $\mathbb{B}(\cdot)'$ on that space such that$$\frac{S_n' - \mathbb{B}'(n)}{n^{1/p}} \to 0 \,\,a.s.$$
Now my question is: What can we infer from that on the sequence $S_n$? Is the (almost sure) limit behavior of $S_n$ and $S_n'$ indeed the same, as the whole sequence has the same distribution?
The statement looks significantly weaker than if we would have something like $$\frac{S_n - \mathbb{B}(n)}{n^{1/p}} \to 0 \,\,a.s.$$ for a Brownian motion on $(\Omega, \mathcal{A}, \mathbb{P})$.
The sequence $\{S_n\}$ has the same almost sure properties as the sequence $\{S_n'\}$ and more generally for any measurable set $A$ in sequence space the chance that $\{S_n\}$ is in $A$ equals the chance that $\{S_n'\}$ is in $A$. The last statement in your post is not significantly stronger than the previous one. Indeed, if the original probability space is rich enough, then the last statement will hold. If not, one might need to enlarge the probability space, but this has no effect on the behavior of the sequence $\{S_n\}$.