in classical Soxhlett extraction, extraction is typically carried out for 24 hours under reflux with solvent quantities of typically 250 ml. This relatively simple work step costs a lot of time and high costs. Microwave assisted solvent extration is much faster (< 30 min) and you need less organic solvent. Acid digestions can also be carried out (e. g. for elemental analysis). However, it should be noted that solvents without a dipole moment (e. g. hydrocarbons) cannot be heated by means of a microwave. This requires small graphite beds melted into glass, which indirectly heat up the solvent.
Microwave assisted extraction (MAE) provides fast energy transfer and simultaneous heating of the whole 'solvent and plant matrix'. By absorbing the energy of the microwaves, the water present in the vegetal matrix promotes the rupture of the cells thus facilitating the release of the chemicals of the matrix and improving their extraction.
From my experience, at shorter extraction time MAE can give higher extraction yield and higher extraction selectivity compared to UAE. To my knowledge for phenolic this applies for sure.