SmartThings allows you to go above and beyond what the ecobee 3 can do just on its own. You'll have the ability to change the temperature, view alerts, and change modes all from the SmartThings app. The custom ecobee 3 device type within SmartThings allows you to take total control of your thermostat. All of these can be connected to the ecobee 3 (or any SmartThings-compatible thermostat) with ease, to perform any number of actions. I have various motion & door sensors, a few presence sensors, water sensors, and door locks in my house. The possibilities for using the ecobee 3 with your SmartThings system really just depends on your actual setup. Sounds like a lot, but it's simple, and works like magic. For me, I use my ecobee 3 temperature to talk to SmartThings via IFTTT and trigger an external switch for a window AC unit. There truly are a ton of ways to use your thermostat with SmartThings, but it really comes down to what other devices you have on your system, and just what exactly you want to accomplish. Once you have your ecobee 3 integrated into your SmartThings setup, the fun begins.
I did all these steps with no issues, and the SmartThings app found my thermostat without a hitch. The basis is to add the custom ecobee 3 device type, then use a smartapp on your mobile device to get it all set up. The instructions there are very thorough, so you should have no problems getting up and running for the first time.
If you don't know much about custom devices or apps - not to worry. That means you'll need to use a custom device type and smartapp, both of which can be found here. (The platform does support older ecobee devices, just not the ecobee 3, yet). The biggest thing to note here is that while the ecobee 3 works great with SmartThings (I've been using one for a few months) it's still not officially supported. Using the ecobee 3 with SmartThings, you have the ability to take total control over your heating and cooling beyond just what the thermostat itself can do, and also control the temperature in your home based on actions of other devices like door/window sensors, motion, switches, or even presence sensors.