Short answer – use the smallest hose (as measured by inside diameter, ID) which will let you get the job done.
The main benefit of ½” hose is that it can put down a lot more chemical. The main benefit of 3/8” hose is that it is lighter and cheaper. Lighter weight matters a lot, especially after 10 or 12 stops in the hot sun.
We have a client that has us put ¼” hose on their 12-volt electric power sprayers. They report that it works fine but we don’t have enough long term data to accurately report on the long term impact of smaller hose has on the pump. The smaller the ID of the hose, the more resistance or friction. The more friction, the harder the pump has to work. Also, longer hoses have more friction than the same size shorter hose.
Most general pest control and post-emergent weed control applications can get away with 3/8” hose. The main reason for this is the spray gun and tips most often used are a few gallons per minute (GPM) or less. This volume is easily handled by 3/8” hose.
Larger jobs, such as large weed pre-emergent jobs or termite pre-treats require larger hoses in order to get the higher volume down faster.