A place to store stuff that can't go in root cellars etc.

[Angelhearth]

I'm trying to organize the storage system, but it seems without an ordinary Storage Barn, some stuff won't be stored (like clothes and tools). A Storage Barn seems to divert the goods away from where it's intended to go, like Root Cellars, Meat Lockers, and Grain Silos, and thus overfill in a short time, making storing of said clothes and tools impossible, which makes the villagers ending up without tools and clothes.

[Vrayna]

In the Storage Cellars section, you have small cellars for just about any kind of good. In the Specialized Storage Barns part, the Crafted/Forged/Fabrics/Misc barns should work to get just about anything that doesn't fit in your special food storage places.

Generally speaking though, things tend to get a bit clunky when using specialized storage. As you've noticed, people seem to prefer using general over special storage. I've never tried building a town with only specialized storage, but I'd expect small issues now and then with it: people might leave work with things they've crafted still in their inventory because they got hungry, go home to eat and get reassigned to a different job while still having their previous stuff. They won't be able to drop that off at the closest barn to free inventory space to start working, they'll need to go all the way to the correct storage to drop the stuff.

If your aim with the special storage is to keep grain near the grain fields and near the windmill for example, I'd use general storage instead but add a grain market stall right next to the windmill. That way the miller will grab the grain from the market and go to work directly, and the stall's vendor will walk around as needed to bring more grain to his stall.
The small market stalls/carts are pretty neat things to the goods you need where you need themn without having to worry about having to worry about what can be stored where and distances for general workers to find the right place to drop stuff.

[Angelhearth]

My reason is to get a good overview of things, and reduce the number of storage facilities needed. Just erecting barns all over the place makes things a bit chaotic. As of right now, the only thing I seem to have a problem with is honey, but there might be others I've just not seen.

To the developer, the cellars are great, but seriously, way tooooo smal. A normal cellar should hold much more than 50 items, just think of your own cellar, and how much you could store there if you only used it to store stuff. What would be great was that each production facility had storage for finished items as well as those needed for production, and in large numbers (say 250 of each). 

[Vrayna]

Yeah, the cellars are tiny, that's why there are specialized barns

There's also a couple of specialized ones in the Quay set of buildings, I always forget about those.

[Angelhearth]

Yeah, the cellars are tiny, that's why there are specialized barns

There's also a couple of specialized ones in the Quay set of buildings, I always forget about those.

But the barn isn't usable, if you want control of what's happening, and don't want barns placed all over your town. 

[Vrayna]

I meant the specialized barns for textiles / crafted / forged / etc etc things.

[stiles]

try building a market to keep the town supplied.

[Angelhearth]

try building a market to keep the town supplied.

That's one of the first buildings I place, after the village has gotten a stable flow of food and firewood. 

[stiles]

after that, you want a mix of different storage barns. do not delete the original pile. thou it can be moved by making a new 1 it is better to leave it alone. there are market carts that will help move materials for the blacksmith and wood cutter,etc. not all modded markets will store the same items. some are better at not storing all limit flags. location of barns matters to outputs. closer the barn, the more production you will get.

you have many many options. each player has to figure out what works best for them. it takes a little while but you will develop your own play style.