If the unit normal distribution is the distribution under discussion, there's no difference. In that case they become the same. Here's how a physicist computes a p-value given a sigma value:
http://i.imgur.com/pcjr6cN.gif
Note that the unit normal distribution is the underlying context.
In statistics, σ refers to an area under the normal distribution defined in terms of standard deviations (1σ = 1 standard deviation).
In statistics and probability theory, the standard deviation (SD) (represented by the Greek letter sigma, σ) shows how much variation or dispersion from the average exists.
I think you're missing the difference.