First, a small correction: Fe, Co and Ni and considered siderophile. Mg, Sr and Ba are lithophile. Ar is atmophile.
Here's one version of the classification, just in case you need to have it in front of your eyes: source: the Wikipedia article on Goldschmidt classification
What is the exact difference between them? It just seems to be the
same thing as sulfide liquid must contain metals as well.
This is a bit confusing because most siderophile metals are also chalcophile or even lithophile. Let's begin by defining what are metal and sulfide liquids in the Earth.
Metal liquid: Siderophile elements are concentrated in molten Fe. Not just any metal, but specifically molten Fe. This is truly molten metal, where most of the atoms are neutral, the bonding is metallic, and there are no ionic bonds. The only place that a liquid like this exists to any significant proportion today is Earth's outer core.
Sulfide liquid: Chalcophile elements are concentrated in molten sulfide. In the Earth this will mostly be Fe-rich sulfide, the liquid analogue of pyrrhotite and pentlandite (known as mss in high temperature). Sulfide liquids can also form from crystallizing magmas when the liquid phase becomes saturated and a separate, immiscible sulfide liquid precipitates. This precipitated liquid too is likely to contain iron as well as chalcophile metals. This liquid is dominated by the presence of sulfide as S2-. This is no longer a metallic liquid with neutral atoms. This is a liquid where the bonding character is a combination of ionic (because you got Fe2+, Ni2+ and S2-), metallic and covalent. Sulfide liquids can exist in the core (we do not know), but they exist in the mantle and the crust where they are how you move the chalcophile (and siderophile) elements around. Ore deposits of these elements almost always form from sulfide liquids, or their alteration products.
Because of the contrasting bonding properties of the metallic and sulfide liquids, trace elements partition differently to them. For example, Te and Cu will "feel more comfortable" in sulfide liquid, whereas Ir and Au will prefer the liquid Fe metal.
How does this fit in the bigger picture? The main importance of the siderophile-element category is that there are not enough of them. When you look at how much siderophile elements there are in the solar system, and then compare to how much we have in the crust and mantle, there are not enough. This is especially true for the highly siderophile elements (HSE: Au, Re, Ru, Rh, Pd, Os, Ir, Pt). This is unfortunate, because these are very useful metals. We could all use some more Au, Pt and Ir. The accepted explanation is that when the Earth formed, the liquid Fe metal that sunk to the core scavenged the siderophile elements from the surrounding silicate (lithophile-nearing) magma ocean and locked them in the core, far away from us.
But this occurred a very long time ago, and has little relevance for moving the siderophile elements around in more recent times, and in places closer to us. In the absence of liquid metal, siderophile elements actually behave as chalcophiles or lithophiles. Ore deposits of HSE and particularly the platinum group elements (Ru, Rh, Pd, Os, Ir, Pt) occur in sulfides. Iron, the namesake of the siderophiles, commonly occurs as oxide minerals or as a component of mafic silicates, giving it a lithophile character.