Sharpening and contrast enhancement are related. The difference is, while contrast enhancement is a global operation (ie., it is applied to the whole image, though it can of course also be restricted to areas of interest), sharpening is a local operation which affects borders and edges.
Sharpening increases the contrast at sharp transitions of brightness, ie. edges, to make them stand out more, which is perceived as increased sharpness. Parameters to choose are the range of edges to be affected (how big the difference has to be), the amount of increase/decrease of tonal values to apply, and the radius of falloff this effect should have.
Contrast is increased by applying an S-curve to the curve diagram, ie. bright values will get to be even brighter and dark values even darker.
There's even a kind of twitter by using sharpening to achieve a local contrast enhancement: unsharp masking with a low amount but large radius (in the hundreds of pixels typically) produces an interesting "clarity" effect.