Well, by definition interpolated means to create a point by using data from the pixels that surround that point. In a picture this would give you higher resolution from an original photograph.
Extrapolation is creating a point by using information contained in pixels which are not surrounding that point.
If * is the new point and + are the originals, then it would look something like this:
Plain:
+ + + + +
Interpolated:
+*+*+*+
Extrapolated:
*+*+*+*+*
Interpolation gives you a high quality, high res image while extrapolation gives a higher res image at the cost of quality.
This is because it has to guess on some points where interpolation has plenty of data to work with. The math in the good programs is insane, so i'm not even going to try and explain it. Bottom line is that interpolation gives the highest quality, but it also depends on the quality of the software.
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"Empirically observed covariation is a necessary but not sufficient condition for causality" - Edward Tufte
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