While both Compost and Compost Tea are beneficial for your yard, there are a few differences in them you want to be aware of when deciding which to apply. 

Dry compost work as a top dressing or when mixed in with soil and contains up to 50% organic matter. This is a slow acting fertilizer and it may take a few days or weeks before you start seeing lusher, greener grass. This is becaue the soil organims need time to digest the organic material and smaller organisms in the compost before the grass can benefit from the nutrients. 

Compost tea is created by suspending the micronutrients of a dry compost in a liquid. This makes for a quicker delivery system of nutrients to your grass. Since the nutrients are in liquid form, the soil organisms benefit from the compost quicker leading to a healthier lawn faster. Compost Tea can also help speed up soil's diegestion of organic fertilizers andsoil amendments. Another added benefit, studies have shown that it may act as an inoculant against some insects and diseases that attack grass.

Both are equally beneficial for your grass, it just depends on how quickly your lawn needs the nutrients!