That is an old thermostat. I have never seen a thermostat with soldered connections! Is the HVAC system the same age as the thermostat? If it is I would love to see pictures of that too. If you have heating and cooling, and you can control the fan, then you need at least 4 wires (Power, heating, cooling, and fan). If you have independent transformers for heating and cooling, then you need two power connections which makes 5 wires. It is difficult to tell how this thermostat is connected. Do you have any documentation? Can you trace the wires back to the furnace and AC? At the very least you will have to figure out the function of each wire. See if the terminals are marked at either end. I don't think the color of the wire is going to help. Most WiFi thermostats required a ground connection. I can't tell if this thermostat has one or not. I am not aware of any WiFi thermostats selling for $100.

I climb up the attic open the door at furnace and find out there is a transformer that step down 120 v to 24 v . and two terminals at the second end (24v) red terminal is power 24v and the other terminal is use to connect to frame or chassis and serve as common ground so a c wire is any unused wire you can pick any color you want in the same bundle that goes from the furnace to the thermostat in my case those extra wires are trim flush to the bundle jacket on both end so I need to peel that jacket off about 2 inches on both end and connect a single wire about 2 foot long from the frame to the wire you selected (let say black) on the furnace end then the black wire on the thermostat end will be "C" wire and connect it to the "C" terminal on the new wifi fancy Honeywell or Nest. If I say anything wrong please correct me. I will do this at some other time. for now I just thinking. I signed up for smart (something) at gapower.com and they will sent me a nest gen.3 for free. I think this nest also will need "C" wire. I wait till I get the Nest then I will install both the Nest and the Honeywell at same time I will put the Nest down stair and Honeywell up stair.

The A, B, C, & D terminals are digital communication terminals that communicate both ways to all equipment connected on this 'network'. If you don't use those terminals (with the proper t-stat) the system can not communicate. There are preprogrammed algorithmic sequences within the equipment boards & thermostat that optimize the HVAC equipment performance given various parameters as to the temp outside / temp inside / desired cooling temp / desired humidity levels. Because the system communicates it knows indoor temp / outdoor temp and possibly even more than that, depending on what was included by the manufacturer. (some equipment now is smarter in the sense it doesn't need the communicating attributes in order to perform, because it performs indirectly by controlling refrigerant temp / pressure attributes. [inverter system]) S1/ S2 and in some cases S3 / S4 are sensor wire for optional attached sensors. If you don't have sensors these do nothing. There are also options to set these as alerts (dry contact) in which they alert you to problems with the system, such as condensate overflow pan has water. These offer no communication attributes. They require a hard wire to whatever sensor they are attached to. Which means you pull more wire. For that reason they are rarely ever used. I am a Texas Licensed HVAC contractor that serves the Katy, Texas area.

What do you mean by "the wires don't match". What type of heating and cooling system do you have? The Nest will work well with some systems, and poorly with others. If you don't have a C (ground) wire then you may have a problem getting the Nest to work reliably.