Since dæmons are effectively the outward souls of humans, their differences will be similar to those of adults vs children.
The dæmon will be more knowledgeable, and perhaps tempered by their years.
They may have romantic affection towards another dæmon, as their human partner would.
They can withstand the touch of another human without revulsion, although this ties to the affection above, but between their humans, as seen with Will and Lyra
They no longer funnel dust into their human, as that was a pre-pubescent phenomenon--as evidenced by the photographic plates, the purpose of the Gobblers, and the settling of Dust.
Otherwise, no strong differences in their function or abilities is given.
The reason why Lord Asriel's Stelmaria was able to defeat two other dæmons is because of two factors:
- She's a snow leopard, a huge predatory cat.
- She's been a snow leopard for decades. This means she's had plenty of experience mastering that form and it's abilities and developing prowess. Based on what we know of Lord Asriel, it's likely she had plenty of opportunities to fight other dæmons, which means practice.
As you see, Pan's fighting ability is a bit limited, due to his unfamiliarity with the full capabilities and limits of other animals. He's spent a lot of time as ferret-like animals (polecat and ermine, nota bly), which is also similar to the pine marten he settled on. That type of animal isn't much use against a snow leopard, so his experience wasn't terribly useful.