Following Basara's line of thought, the mining skill should include the ability to refine ore into ingots. But Smthing should allow a smith to re-smelt or melt items that are already made from refined metal. I like the idea that having mining would increase the amount of ingots received from smelting.
As Basara also alluded to, in days of old, a Smith would typically only work with 1 metal or 1 category of metals...
Blacksmith: Iron or iron alloys such as steel. The term "Blacksmith" is based on the dark or black color of iron when it is in ingot or cold form.
Whitesmith: worked with white metals such as tin or pewter.
Coppersmith: worked with copper and its alloys (Brass, bronze). With the discovery of iron, most Coppersmiths switched to Blacksmithng.
Silversmith: worked with Silver (Paul Revere was a Silversmith). Mostly done with cold metal. Forks, spoons, knives, plates, teapots, jewelry.
Goldsmith: Gold, of course, They also worked with other precious metals including platinum, copper, and silver. In fact most Silversmiths and Goldsmiths worked with the same metals, but were known by their primary metal (either silver or gold). Both smiths made jewelry which also included precious gems or stones. If they made armor or weapons, they were only show pieces because gold and silver are too soft to protect or clash with other weapons in battle.