The greatest advantage is that you can date the tectonic fabric and multiple periods of metamorphism with one sample. However you must identify where in the thin section the monazite is growing and you must know the chemistry of the monazite. Monazite can grow in the plane of the foliation, whereas zircon does not do this very often. For example if monazite is growing at the same time as garnet, it will have different trace element concentrations than if garnet is breaking down. Monazite is great in dating polydeformed terranes.
However, zircon ages may be more reliable. With monazite dating you must assume that the monazite has not lost any Pb.
In my opinion, both minerals have their pros and cons. Both minerals grow under different temperatures forming different grain morphologies. Shape difference which can be driven down to the crystallographic level by labeling each face and edge with the Miller´s indices under SEM is diagnostic of the physical-chemical regime and particularly helpful at the brink of forming pegmatoids or aploid mobilizates. While zircon is almost inert to hypogene and supergene fluids monazite is not and as a LREE phosphate decomposed into florencite (pH < 7) or rhabdophane (pH >7) so that you can find sometimes one or the other reworked into residues such as stream sediments or lithified within sandstones.
For further reading have a look at
DILL, H.G., WEBER, B. and KLOSA, D. (2012) Crystal morphology and mineral chemistry of monazite–zircon mineral assemblages in continental placer deposits (SE Germany): Ore guide and provenance marker.- Journal of Geochemical Exploration, 112: 322-346.
You have some very good answers above. The different crystallisation temperatures allow you to investigate different aspects of the P-T history. Zircon crystallises at very high temperature, so can tell you when peak high grade metamorphism occurred. Monazite crystallises at lower temperatures, so can give you the timing of peak moderate grade metamorphism. With high grade metamorphism, monazite might tell you when the rocks were heating prior to the peak, or cooling after the peak.