Garden Hose....diameter vs. outlet feed

The pipe leading to my outside water source is 1/2" brass.

If I use a 3/4" garden hose as opposed to 5/8", is there any extra added benefit ? Or is it a cumulation of hose length, water pressure, ID of the hose in relation to ID of the brass pipe, etc ?

Every inch of pipe, tubing or hose contributes to pressure drop and a reduction of flow at the end. Using the 3/4 inch hose will reduce the pressure drop across the length of hose. The longer the hose the more pronounced the pressure drop will be; therefore, using a 3/4 inch hose will have less pressure drop (and consequently a greater flow rate) than using the same length of 5/8 inch hose.

hose

The size of the hose is often immaterial, because the amount of flow is limited to what the orifice in the hose's faucet can deliver. Since that opening is usually a 3/8" hex, or less, that is all the water you will get as long as the hose is large enough to deliver that amount.