The three models I am looking at are the 560 iv, 568ex and 685.
I have been able to determine that the 560 doesn't have TTL, which I'm not sure if that's important or not, while the other two do.
TTL is a way for the camera to automatically set the flash's power based on metering. The camera tells the flash to send out a low-power "preburst" to meter, meters it, and then sets the flash power. It's very useful for run'n'gun event shooting with the flash on the camera, such as parties, weddings, social shooting, etc., where you don't have time to monkey with a flash on a stand and do setups, and where you're always going in and out of different lighting conditions. Think of it like having aperture-priority mode on the camera. It's like ambient-priority mode, with flash-to-fit.
You'll also want to find out about high-speed sync (HSS; aka FP or focal plane flash)--the other big thing the 560s can't do. These two features (TTL and HSS) often go together, because they both require full hotshoe communication.
See: What features should one look for when selecting a flash? and my answer on "How do I correct overexposed fill-flash when shooting outdoors on a sunny day?".
I have also seen that it seems to be really hyped that the 560iv can be used as a wireless master, something I wouldn't use right away but could come in handy later. I don't know if the other two share this functionality.
They do and they don't, depending. But as radio masters, no. The 568EX II is an optical master. The 685 is slave-only, much like the old YN-560 III was and is following the same pattern: first gen, slave-only w/built-in radio.
I haven't seen it mentioned anywhere, but it seems odd that this nice feature would be exclusive to the cheapest of the three models.
It's because it uses a super-cheap radio triggering system (RF-60x/560). The other two don't, and are mostly expected to be used in the YN-622 triggering system, which is completely incompatible with the 60x/560 system.
I should also mention that the 560IV has been succeeded in the Yongnuo lineup by the YN-660, which has a better UI, more groups, and larger zoom range.
I can't really tell if there is any significant differences between the 568 and 685, feature wise. They also seem to be very close in price most places. It would be nice if either one of them had off camera TTL.
The significant difference between the 568EXs and the 685 is that the 685 has a YN-622 receiver (which can do off-camera TTL/HSS) built-in. The 568EXs do not and require adding a YN-622 to the foot. So, the 685 is actually cheaper in this respect. However, because of the 622, Yongnuo decided you don't need optical slaving any more with the 685 and removed those modes. That's why the 685 is not an "EX" model--it can't be used in Canon/Nikon "smart" optical slaving systems, or with the dumb "S1/S2" modes.
See also: What are the Yongnuo flash naming conventions?
Just my opinion, but consider going with Godox instead of Yongnuo (well at this time, that might change in a few years). Messageboards are a little behind the times, and Godox has just released its 2.4GHz X system of lights and triggers. Unlike Yongnuo's gear, they're designed as a system, with a lot more capability for expansion. It also doesn't cost much more at the low end than Yongnuo.
I'd say a Godox TT600/V850II (vs. a YN-560/660) or TT685/V860II (vs. a YN-685) and an X transmitter are a better buy. Because you can mix and combine TTL and manual gear in the system (same triggers for both), and have the option to add bare bulb flashes and studio strobes both manual and TTL. All of which have built-in triggers and HSS capability. And the AD600 has features you can't get on an Einstein (Bowens mount, ability to separate into a pack'n'head), battery-powered, TTL, HSS, built-in radio slave, etc). Oh, and the system supports Canon/Nikon/Sony/Fuji/MFT and the TTL gear (except the 350 models and the X1R receivers) can autoswitch for the different brands. So, if you ever jump systems or add mirrorless gear, you only need to rebuy anything that goes on the camera; not any of the off-camera lights. Not so much the case with Yongnuo.