The most jarring change is that the weapon triangle is essentially gone. There are still swords, axes, and lances, as well as tomes, bows, and a new weapon in gauntlets. Each can still have contextual strengths, like bows being particularly powerful against flying units and lances against cavalry. But the broad rock-paper-scissors triangle is gone, instead replaced by a deeper Combat Arts system.
While a sword won't naturally beat an axe, teaching a character the Axebreaker combat art will give them an edge in battle. These arts are powerful moves and abilities that can go above and beyond the average attack, but cost additional weapon durability. There is the tradeoff: do you want to hit an enemy with everything you've got now, knowing an attack three turns later might break your lance and leave you without a weapon?
You develop these arts, as well as each character's proficiency, talents, and support relationships in the school-life sessions. While Fire Emblem has typically been broken into chapters centered around key battles in the past, Three Houses' chapters are much larger in scope. A single chapter will be a calendar month, where you spend the week instructing the students of your chosen house and training them for combat.