According to Wikipedia, strict partial orders and strict weak orders are transitive binary relations satisfying asymmetry, antisymmetry and irreflexivity. See screenshot below:
However, if strict partial orders and strict weak orders have different rows in the table, it must be because they are somehow different. Hence,
What is the difference between a strict partial order and a strict weak order?
Also, can you provide examples that illuminate the difference?
Without any qualifiers the term partial order typically means a relation on a set $X$ that is reflexive, antisymmetric, and transitive. An example is $\subseteq$ on the power set $2^S$ of a set $S$. The reflexive reduction of a partial order is called a strict partial order. The analogous example is $\subsetneq$ on $2^S$.
A partial order is a weak order if it contains no ${\mathbf 2}+{\mathbf 1}$, in other words, if no distinct elements both incomparable to any third distinct element are comparable to one another. An example is the has-lesser-cardinality-than-or-is relation $\preceq$ on the power set $2^S$ of a finite set $S$. Its reflexive reduction is a strict weak order. The analogous example is has-lesser-cardinality-than $\prec$ on $2^S$. We have $\{3,7\} \preceq \{1,2,4\}$ and $\{3,7\} \preceq \{3,7\}$, but $\{3,7\}$ and $\{10,20\}$ are incomparable. By contrast, under $\prec$, $\{3,7\}$ is incomparable to itself.
Wikipedia's weak ordering page contains this quote:
A strict partial order $<$ is a strict weak ordering if and only if incomparability with respect to $<$ is an equivalence relation.
So it seems that every strict weak order is a strict partial order, but not conversely. For example, consider the relation on the set $\{1,2,3,4,{\between}\}$ where $1<2<3<4$ and $1<{\between}<4$ but neither $2$ nor $3$ is comparable to $\between$. This is a strict partial order, but it's not a strict weak order: $2$ is not comparable to $\between$, and $\between$ is not comparable to $3$, but $2$ is comparable to $3$. (Actually $1$ and $4$ aren't even needed for a more minimal example.) If this order were changed so that $2$ and $3$ were not comparable, then it would be a strict weak order.
