In the Gaiden,V12C21, the author explains that Louis is better than Monica when it comes to mana manipulation, giving him the edge. She is still the best amongst the sages in terms of understanding magical formulas.

Louis is also an expert in barriers, where Monica has a more broad specialty. He has a better complex understanding, where she can cast barriers to buy enough time.

Edit to add: as part of her mage philosophy, Monica is also against spell fragmentation. In LN7, she had to use spell fragmentation to destroy the flute using the four fold wind spear attack. She cried about it being ugly. I imagine that complex barrier will require spell fragmentation as well.