This is the fault line the whole field is organised around, and it is worth stating why it is so stubborn: each position is strongest exactly where the other is weakest. Dualism honours the datum that experience seems irreducible; physicalism honours the datum that minds depend utterly on brains. Neither datum is going away.
Dualism's case begins with an observation nobody disputes: describing a brain state completely does not seem to describe what it is like to be in it. You could specify every neuron involved in tasting coffee and still not have conveyed the taste. Dualists take that appearance at face value — if the physical description leaves something out, then something is not physical. Descartes' substance dualism is the historical version; most contemporary dualists prefer property dualism, keeping one substance while insisting some of its properties are irreducibly mental.
The cost is interaction, and it is heavy. Physical events have physical causes, and the physical world appears causally closed — there is no unexplained shove in the motor cortex waiting for a soul to supply it. If the mind is non-physical yet moves the body, either physics is incomplete in a way no experiment has detected, or the mind does not really do anything, which most dualists are unwilling to accept.
Physicalism's case is the overwhelming evidence of dependence. Alter the brain and you alter the mind, reliably and specifically — anaesthesia, injury, and pharmacology all demonstrate it. That the mind depends on the brain is not seriously in doubt, and physicalism gives the simplest explanation: it depends on it because it is it.
Its cost is the explanatory gap. Granting a complete neuroscience, the question "but why is there something it is like to undergo that?" seems to remain askable — which is what Mary's Room and the zombie argument are constructed to expose. Physicalists split on how to answer. Some argue the gap is epistemic, a limit on our concepts rather than on reality. Some, following Dennett, argue the intuition is an illusion produced by unreliable introspection. Others accept the gap and adopt property dualism or panpsychism, which is really a concession that the strict version does not work.
A note on how to weigh this. It is tempting to treat physicalism as the default because science is successful, and dualism as the default because experience is undeniable. Both moves beg the question. The productive framing is the trade: dualism preserves the phenomenon and struggles with causation; physicalism preserves causation and struggles with the phenomenon. Which cost you find more tolerable is, at present, where the argument actually terminates.