New ecobee3 lite, have heat but no cool option
You skipped the compressor contactor
When you were reading your furnace's wiring diagram, you skipped over the symbol between the thermostat Y terminal and the furnace C connection that represents the coil on your air conditioner's compressor contactor.  That coil is what is energized by the thermostat to turn the air conditioning on, so it's kind of important that it not be left out of the circuit, as you seem to have assumed so far.
Fortunately, you can fix this rather readily since you have a spare wire handy from the looks of things.  Strip the end of the spare (blue) wire at the furnace end, and join it in with the junction of the fat brown and thin red wires (by the wiring diagram, that's your C wire connection); then you can wire yellow to Y and blue to C at the thermostat end.  From there, finish wiring the thermostat as you had it before, button everything back up, and enjoy your new thermostat!