Metal vs. Styrofoam cup calorimeter?
You're right that IF the metal calorimeter and the styrofoam calorimeter were both perfect and behaved in the way that we usually pretend calorimeters do (they don't absorb or lose heat) then they should produce identical results, for the enthalpy reason you provide.

However, since we're asking about specific materials, metal vs styrofoam, we're considering real calorimeters instead of ideal.

There are a few things that calorimeters can do that will affect how large a temperature change you measure.


Heat loss - As you note, metal conducts heat much more effectively than styrofoam.  This means that the metal calorimeter will carry the heat of the reaction away from the reaction vessel to the outside world faster than styrofoam will.  If that is the case, which reaction vessel will get hotter, styrofoam or metal?
Heat absorption - The calorimeter (and thermometer) gets heated up along with the reaction mixture.  A calorimeter that absorbs large amounts of energy will have a relatively lower final temperature, since more of the reaction's heat went into heating up the calorimeter than went into heating up the reaction mixture.  This quantity is known as the calorimeter constant and varies depending on the materials and mass of the calorimeter.  Styrofoam absorbs more heat per gram than steel, but the steel calorimeter is probably heavier.  With a fast reaction there is the additional fact that the styrofoam calorimeter probably never comes up to the full temperature of the reaction before you quit recording temperatures.  This winds up leading to a small source of lost heat for the metal calorimeter as compared to the styrofoam.
The quality of the equipment -  Metal calorimeters are often made specifically for the purpose of calorimetry.  Styrofoam calorimeters are often a couple of coffee cups jammed together.  If the calorimeter allows a significant amount of heat loss to the atmosphere due to its construction, then it will result in a cooler reaction mixture and a lower calculated enthalpy for the reaction.


You've got to take all three of these into consideration in order to determine the particulars for your experiment.