I think you could simplify the question if you focus on the above the ground and underground biomass. I assume strong positive correlation between the biomass and water absorption capacity. There are other important factors to consider (rain intensity, soil composition, slope, etc.).
Agree there are other factors. Many of the US Forest Service hydrological stations have studies addressing. For equal soil types and depths, where both grass and forests can grow, forests tend to use more water in evapotranspiration, so due to this fact alone, the forest soils would have a greater capacity to absorb more water, as soil moisture on average would be lower. In addition, it is fairly well accepted that well developed forest soils have extremely high infiltration capacity, with minimal runoff. Certainly there can be some exceptions, such as severely eroded and gullied lands in forest or grassland with poor soil development, and conditions where the dense rooting capacity of grasses may be beneficial to some soils where fine sediments have clogged macropores. In NC, USA at Coweeta Hydrological Station for example, studies in the Blue Ridge Mountains have shown water yield from grasslands to be higher than for hardwood forests, which are both higher than water yield from pine forests. Pine use more water in transpiration, and their stomata are open longer during growing season and do not close as tightly. Since pine forests use more water and deliver less streamflow, their ability to take up water into their soils must be Enhanced. I hope you can appreciate the logic. This does not mean there can be exceptions. Not all soils, vegetation respond the same. When working with eroded and gullied lands, we have seen great success in stabilizing landscapes with converting the eroded farmlands back to pine forests, resulting in substantial reduction in water yield and erosion. We do use a combination of annual grasses for rapid cover, native plants and planting pine trees, sometimes needing fertilization and mulch in the poor lands. In recovery of damaged soils, grasslands with dense root systems such as switchgrass can probably develop organic and absorption capacity at a greater rate than forests, that tend to be slower growing. Even though some invasive species have been used in the past in recovery efforts, we need to avoid using them, and find acceptable solutions. Hibbart (~1962) compared many international water yield studies from forests and grasslands, you might look up this publication as an example. Since there are quite a few hydrological research stations in USA, you might select one most similar to topography, vegetation, rainfall, soils, etc. and review their publication list for applicability. Also Hibbart and other water yield researchers typically have some excellent citations to review.
Confirming William F. Hansen answer, and for example depended only to the surface runoff and infiltration rate relation , we can say that the ability to absorb water within the “Forest with good vegetation on forest floor”(high infiltration rate with high residence time of water in contact with plant root) is higher than district of “Grass land”, but this case is a primitive prediction and not always represents the fact.
Like it has been said, the ability of any vegetation type to absorb water is dependent on factors such as soil type, topography, rainfall intensity and frequency and sometimes on geomorphology. High interception is expected in forest area than grass land and this have impact on access to the soil. Then type of soil and it's composition determines the rate of infiltration. For example, the behaviour of clay dominated soil even in forest area 'll be totally different from what happens to a Sandy loam dominated soil in a grassland. More water will be absorbed in the later than the former. The relationship continues like that. So, the characteristic of d soil, topography, geomorphology of case study you re looking at 'll determine the rate of absorption and runoff.
This depends on many factors, such as gradient, soil texture, soil depth, rainfall intensity, climate, and so on. For example, the performance of different vegetation cover varies from different slopes.
Agree there are too many integral factors for a complete answer. Add soil biologic activity, soil structure and water-stable aggregate quality as factors.
There are many factors to adjust. Typically, forests with different roots depths in soil and different layers of canopy tend to absorb and store more water.