Compressor/fan stays off when cooling turned on
The first thing to do is check the two wires on the outside unit when the thermostat is calling for cooling.  In your picture it looks like the Blue and Yellow thermostat wires are being used.  You should have 24v AC across those two wires, which would normally close the 240v relay and start up the compressor and cooling fan.

If you're not getting 24v on the two thermostat wires, there's probably a bad connection somewhere.  Since the air handler is working and the thermostat has power, the 24v transformer is working.

If you are getting 24v, then you can move on to diagnosing just the compressor.  There are limit switches that can prevent the compressor from turning on if the coolant level is off.  There can also be temperature sensors that prevent the compressor from running, so these would need to be checked.  The relay should also snap shut when the 24v signal is present.  A poor connection might read 24v, but might not have the amperage to actually close the relay.  The relay can be corroded or filled with dead ant bodies.

I hope this helps.  Clarify anything you can in the comments and I can attempt to edit as needed.  I've done a lot to diagnose and solve problems with a WiFi thermostat and heat pump that were not wired correctly.