There should be no problem connecting the C wire from your thermostat to the C terminal in the furnace. Take a look at this diagram, which is a rough approximation of your system.
Notice the cable going to the condensing unit has a red wire connected to the Y terminal in the furnace, and a white wire attached to the C terminal. When the thermostat calls for COOL, it does so by energizing the Y terminal (connecting R to Y). This provides power to the coil of the contactor, which allows it to close the contacts and turn the condensing unit on. The other side of the contactor coil is connected to the C terminal, as to provide a complete circuit.
You'll notice in the first image that two relays in the furnace, the thermostat, and the contactor coil all connect to the same leg of the transformer. This is because the thermostat basically works like a switch, connecting the R wire to the C wire through control circuitry. Anything that needs 24VAC power, will be connected to both legs of the transformer.
This is my attempt to "demystify" the whole "C" wire thing for non-technical folks. This explanation intentionally ignores lots of details that are not relevant to a basic understanding of the concept.
An old-timey thermostat is mechanical in nature. You can think of it very much like 3 separate mechanical on/off switches. (the "switches" in this case are mechanically actuated by temperature using springs and tilting glass bulbs full of mercury and other old-timey mechanical mechanisms)
The circuit is actually VERY simple. The old-timey thermostat switches on and off the "juice" which comes in on the RED wire...separately on and off to each of the other 3 wires (often green for the fan, white for the heat, and yellow for the AC).
That's all the old-timey thermostat does. Hook or not hook the red wire to each of the other three as needed.
Modern thermostats control your HVAC system with the same connections. They do the same thing...hook the red wire up to the proper green/white/yellow to turn various parts of you HVAC on and off as needed.
Since the old-timey thermostats were mechanical in nature, they didn't need any juice of their own to operate. If you think of a modern thermostat as an old-timey thermostat plus a smart phone stuck to the wall right next to it...you need something to power the smartphone. (some "kind of modern" thermostats have batteries in them for this purpose...to power the "smarts")
The red wire brings juice into the thermostat...but there is no "ground wire". Our "smartphone stuck on the wall next to the old-timey thermostat" needs both a "hot wire" and a "ground" (or "common"). (again, this is simplified so as to not get lost in irrelevant details and confuse the non-technical readers this was written for)
The "C Wire" (often blue) is this additional "ground wire" needed in order to complete a circuit for the electronics (which old-timey thermostats didn't have). The "other end" of the C wire is hooked to the other side of the transformer that supplies juice to the R or red wire. It is the other wire needed by the electronics in the newer thermostats in order to power up their "smarts".
Hope this helps a few people understand what the C wire is about.
Best Answer
I just got off the phone with Honeywell. If you only have a 4-wire older system, you'll only be able to use the fan in the "auto" position. It won't let you use the fan-only selection. You need a 5th wire for power for the wi-fi side of the thermostat. Call Honeywell and they'll walk you through it for free.