How are these two different or are they the same thing? if they're different, which one is better for what? What's the pros and cons of each?
They are different, but often associated together in practice. A sump is basically where equipment like a protein skimmer, heaters, media reactors, etc. are kept so that they are out of view and not cluttering up the main tank. It also acts as a way to gain total water volume and increase overall system stability. Often, live rock is placed in a sump for extra biofiltration. A refugium, on the other hand, is generally where macroalgae is grown, is lit, and considered a place set aside for copepods / amphipods / mysid shrimp to grow. In many cases, a refugium is set up as a section of a sump, but there are cases where each is completely separate from the other.