Pretty much all commands break, since the entire command system was overhauled. With some automatic converters you can upgrade them or alternatively replace them manually with better solutions that are new in 1.13. Most changes can be found here (archive), but that post was made before the changes were done and there are some additional changes. But the new autocomplete is very good and mostly explains itself and the wiki was over time also updated, it should be accurate again now.
The data pack structure has changed and of course texture packs also have to be updated to include the new blocks, entities, etc.
Customised worlds don't work in 1.13. They can't be created, but they can be opened. When opened in 1.13, any new chunks generate like in a default world. The "buffet" world type replaces customised worlds, but it's neither completely done yet, nor does it have all features of customised worlds in 1.12. It's planned that they return later, but there's currenty no date or version known for that.
There are a few other world generation changes, they might cause a few visible chunk borders, but nothing major.
Mob systems that drown zombies to kill them don't work anymore, because they are converted into "Drowned" now.
All undead mobs sink in water.
Baby zombies burn in daylight.
You'll have to either wall yourself in or have a solid block above you for AFK farms now, Phantoms spawn when you haven't slept for at least 3 nights in a row. They fly, but unlike vexes and dragons, they don't fly through blocks. Having a solid block above you prevents them from spawning.
Water physics have changed a lot. In theory they shouldn't break existing builds, but there might still be some bugs with edge cases left.
Items and XP orbs float up in water.
Beds break if they have no supporting blocks under them. Since they were non-functional without support before anyway, this should only affect decorative beds.
TNT that explodes when mined was removed and turned into regular TNT in 1.13.0 and readded in 1.13.1. If you skip 1.13.0, they should upgrade properly.
And most importantly: Because there was a deadline for 1.13, it was developed pretty hastily and there were still many bugs, some quite game-breaking. There's also a lot of lag, both fps and tps lag.
In general I would not recommend updating to 1.13 at all. Instead, you should stay in 1.12.2 or, if you want the new features, directly update to the latest 1.14 snapshot.
Usually, snapshots are unstable, buggy, laggy etc., but since 1.13 was in such a bad state, the current snapshots are actually more stable than all 1.13 releases.
Keep in mind that snapshots are development versions. The 1.14 snapshots currently still have some quite severe bugs, even if they aren't as bad and as numerous as in 1.13. For a stable game, you should still stay in 1.12.2.
Of course 1.14 has additional changes, the main ones that impact an existing world are these:
Some commands, datapack, resource pack and similar related things have changed again, but not as drastically as with 1.13.
Sticks are dropped by leaves sometimes, so if your tree farm assumes that you only get saplings and wood, you should consider that.
Dispensers can now shear sheep with shears. I don't really expect this to break anything, but it could in theory.
Ocelots no longer turn into cats when tamed. Instead, cats are their own mob.
Combining tools, weapons and armour in the crafting table no longer works, this task was moved to the new "grindstone" block. Of course you can still use the anvil.
Vindicators can break doors.
The textures were updated to the new pack that was available for a while already. The old textures are available in the resource pack menu.
Lapislazuli, cocoa beans, bone meal and ink saks now have to be crafted into dyes before they can be used as such.
In both 1.13 and 1.14, there were a few changes in world generation (mostly oceans in 1.13 and villages in 1.14), so in some places you might see borders between old and new chunks.
And again, most importantly: There's still something going on at Mojang that we don't know, which causes them to pay WAY too little attention to bugs. So there are still really critical bugs left, including an entity deletion bug, CPU usage of dedicated servers not going down from 100% in certain situations (which makes cloud hosting pretty much impossible) and chunk loading outside of spawn chunks or render distance of players being extremely broken, including
There are hints that this might change in the future, so we should hopefully get a better 1.14.x version at some point, which I'll then recommend updating to. But for now, stay in 1.12.2, if you can.