If comparison is cheap (e.g. when sorting integers), pdqsort wins because it copies less data around and the instructions are less data-dependency-heavy.
If comparison is expensive (e.g. when sorting strings), timsort is usually a tiny bit faster (around 5% or less) because it performs a slightly smaller total number of comparisons.
"stable" is a simplified Timsort: https://github.com/rust-lang/rust/pull/38192
"unstable" is a pdqsort