Traditional sterling silver is an alloy of 925 parts silver to 75 parts (usually) copper. This helps to provide some strength to the metal, as pure silver is quite soft. The copper though, also helps to tarnish the metal. Personally, I love sterling silver that gets some black in the grooves of the piece because it really helps enhance the details but many people like their metal bright and shiny.
Argentium is also an alloy, but one made of silver and germanium. It is resistant to tarnish and cleans up quite nicely, often just in a warm soapy bath. I find if I leave it out and exposed to air it goes a kind of goldish, brassy colour. What is important to me in the difference between these two metals is their applications - what I do with them and how come I choose that particular one for the job.
When using the torch to ball up the little end of the metal for use as headpins, nose jewellery, the little bud earrings I make, or the ball end earwires I often use, I choose which one based on what I want the outcome to be. Argentium makes very smooth, fluid little balls, whereas traditional sterling makes little balls that are slightly pitted, or cratered. In the photos below, the first photo shows earrings made with Argentium, and the second shows traditional sterling - see the difference?
If I'm going for a smooth sleek look, I'll choose the Argentium, but if I want something a little more earthy and organic looking, it's traditional sterling silver hands down every time.