Zigbee has multiple profiles and only some of them will work with smartthings. Short answer is those using the zigbee home automation profile (ZHA 1.2), or the newer Zigbee 3.0 profile. (But not devices which require “touch link commissioning“ or “Zigbee green power” neither of which are supported by the smartthings hub.)
See the profiles FAQ (the topic title is a clickable link)
Hue devices are Zigbee 3.0, but should still be used with the hue bridge.
FAQ: Do I need the Hue Bridge/Hub to Use a Hue Bulb with SmartThings?
If a zwave or zigbee device does work with smartthings, then you will not usually use its own hub/gateway. (Again, Hue is an exception because technically it’s a bridge, not a hub. It forms its own mini Zigbee network and then communicates with other platforms, like smartthings, by LAN or cloud to cloud.) so for example if you get IKEA Tradfri Zigbee Devices, you would connect them directly to a smartthings hub rather than using the IKEA Gateway. The same with Xioami zigbee devices.
Let’s see, what else. 
Z wave “local scene” controllers will not work well with smartthings. There aren’t too many of these around anymore, it’s an older technology. Z wave “central scene“ controllers will probably work with smartthings, but may need custom code.
Most Z wave devices will work at the “basic“ level for on/off/Dim, but may require custom code for more advanced features which are an optional part of the specification.
One more thing: smartthings has never been very good at “multi endpoint“ Devices, whether they are Z wave or zigbee. So if you are looking at multi button devices, always check the forums first to make sure that particular device works unless the manufacturer has specified that the device works with smartthings.
To be honest, it pretty much all comes down to the first rule of home automation: “the model number matters.“
While the smartthings hub has both a zwave radio and a zigbee radio, that doesn’t mean it will work out of the box with every Z wave device or every zigbee device, or that it will have access to all the features of those devices.