There are many smaller and larger differences between Total War: Warhammer and previous entries in the series. This answer will focus on the more important differences which affect how the game is played. Keep in mind that this is a work in progress and may be expanded over time.
In Warhammer, characters can be fitted with equipment, making them much more flexible than in previous installments, as they can equip or unequip different pieces of equipment and even mounts depending on the situation.
There are 3 types of characters in Warhammer, the Lord, the Hero, and the Legendary Lord. Lords and Legendary Lords can be assigned the role of General (land armies), Admiral (naval armies), or Overseer (settlements). Unlike previous Total War games, Lords do not have bodyguards and are instead represented by a single, particularly powerful character.
The Legendary Lord is a unique (named) Lord with a unique skill tree on top of the Lord's usual tree. Unlocking unique skills can unlock unique quests for the Legendary Lord to complete.
Heroes act like agents, in that they can perform actions like assassinations, sabotage, and more. Unlike in previous installments, agents (Heroes) can actually participate in a battle like every other unit, as long as they're assigned to an existing army. Just like Lords, Heroes do not have bodyguard units.
In previous games, you could have multiple generals in an army. In Warhammer, you can not have multiple Lords in the same army. In return, Lords do not always die when they fall in battle. Sometimes, they just end up wounded, and can return after a number of turns. Legendary Lords never die unless their faction is on the brink of annihilation.
Whenever one of your armies win a battle, they may receive (or "steal") an item. These items can be pieces of equipment to wear, scrolls or potions to consume in battle, or banners to assign to a unit. Aside from banners, only Lords or Heroes can use items. Banners can be assigned before or after battle to any unit in an army, to which they grant a buff.
Magic is something that so far only exists in Warhammer. It is very similar to abilities. To use magic, you first need to learn a spell. Casting spells consumes Magic Power, which is shared among all of your units. This means that if one of your mages uses too many spells, your other mages won't be able to use their own spells either until Magic is restored. In return, spells are very powerful.
When a battle starts, you will rarely have a full pool of Magic Power. However, if you feel like you don't have enough starting Magic, you can gamble to get more. If you're lucky you can start the battle with more Magic, and if you're unlucky you'll end up with even less. Gambling can only be performed once per battle.
In previous Total War games, you could only select a faction to play as. Each faction would have different perks, such as better archers or better swordsmen.
In Warhammer, you choose a race instead. Technically, you still get to choose between different factions (the Empire is a human faction, the Dwarfs is a dwarven faction, etc.), but you only get to choose a single faction per playable race. After you have chosen a race, you can select a Legendary Lord to start with; the other Legendary Lords can then be hired after their requirements are met. In addition to having different perks, each faction also has its own Legendary Lords.
Different races have a significantly different play style from one another, with their own unique units and mechanics. For instance, the Empire (humans) is a very traditional race, which can use diplomacy and warfare to their advantage; the Greenskins (orcs & goblins), on the other hand, are a very offensive race that need to keep their armies busy, else their troops will start fighting and killing each other. More information on how the races differ from each other can be found here.
In the past, all faction had the same tech tree to research from. Also, the buildings you could build and the units you could train depended on the technologies you've researched. In Warhammer, things have changed considerably. Each race has its own tech tree; some — like the Empire — don't even have a "tree" so to speak. A number of races — like the Dwarfs — don't even need to research in order to build every building or train every unit. There are races which need to build certain buildings in order to research certain technologies, and others who only need to research the prerequisite.
It is no longer possible to occupy all settlements, as some settlements are only available for certain races and therefore need to be razed by others. For instance, humans and vampires can only occupy other human settlements (except for Norsca) or vampire settlements. Dwarfs and Greenskins can only occupy other Dwarf or Greenskin settlements. Chaos Warriors and Beastmen cannot occupy any settlements.
Warhammer introduces flying mounts for (Legendary) Lords and Heroes, as well as flying units. Flying units have the advantage that they are harder to kill with melee ground units, which means their natural enemies will be ranged units, units that can cast spells, and other flying units. Some flying units can attack from a range and can therefore not be killed by melee ground units.
The naval battles we're accustomed to since Empire are currently not available in Warhammer. Instead, they can only be auto-resolved. According to fan theories, this is because the races currently available do not have any significant water bodies separating them, meaning that naval battles may be added later when more races become playable.