What are the conditions under which one would normally switch on the Engine and the Wing Anti Ice?
In general the cowls get operated way more than the wings, because the operating requirements are more stringent, because ice can form on cowl inlets more readily than the wings in conditions near freezing (temperature drop from inlet flow) and the consequences can be worse when the ice is shed.
Can't speak for the 737, but on the CRJs at least, the cowls have to be on in conditions where the wings don't, such as flying in visible moisture above 230 kts (above which the wings only have to be on if you get an actual detection of ice on the airframe).  So when you depart in real icing or &quot;technical icing&quot; (visible moisture below a certain temperature) conditions, both wings and cowls will be on, but during the enroute climb you don't need the wings on without an ICE caution message, whereas you still have to put the cowls on pretty much any time you are in cloud.
In the CRJ program the duty cycle of the cowl AI system was several times higher than the wing AI system because of this, and I'm sure most airliners are similar.