Well, Supergirl is very straightforward in her actions. She sees a problem and wants to solve it right away using her powers. Batman, though, is more methodical. He plans every move carefully. In a team - up, this could be both a clash and a complement. The clash could come when Supergirl wants to rush into a fight, but Batman needs more time to set up his gadgets and analyze the situation. However, it's a complement because Supergirl's eagerness can push Batman to act faster when necessary, and Batman's methodical approach can help Supergirl be more strategic in her actions. They could also have some great character - development moments as they learn from each other's personalities.
Supergirl is generally more optimistic and believes in the good in people. Batman, on the other hand, has seen the worst of humanity and is often cynical. This could lead to some clashes. But it can also be a great complement. Supergirl's optimism can help Batman see the better side of things sometimes. And Batman's caution can prevent Supergirl from making hasty decisions. In a fanfiction, we could see this play out when they are dealing with a villain who claims to have reformed. Supergirl might be inclined to believe them, while Batman would be looking for signs of deception.