Adjustability, also more fluid so they don’t get as hot so they’ll have a more consistent feel through various conditions.
Apples to apples for the sake of comparison, you can just as easily tune a steel body emulsion without a reservoir as you can one with a reservoir. Typically the only difference is the addition of the reservoir.
The reasons we run RR shocks are for linear shock response over time. The emulsion shock mixes the compressed gas charge that is over the oil with the oil and makes the fluid spongy which varies the shock response over time. There is a slight benefit of heat reduction but not much really. The only additional oil is what is replacing the gas charge but we also typically use that space to fit a longer shock in the same place. Or put another way, a RR shock can be shorter due to eliminating that air space from the body and converting that to piston shaft length.