What is the difference between cardinal vowels and pure vowels?
Cardinal vowels and pure vowels are very different categories of vowels with no overlap in the definitions of the eight primary cardinal vowels and the seven primary pure vowels.
Pure vowels are those few, which may be used in most language, that maintain a steady-state, single vibrational sound frequency produced by unchanging vocal tract shape and tongue/lip position.
Cardinal vowels are those few that are common to all languages and that mark the corners and four equidistant points of the legs of the vowel quadrilateral. They were designated by Daniel Jones in 1956 as the reference vowels for describing the sounds of any language in terms of (1) tongue position high or low, (2) tongue position forward or back, and (3) lip roundedness (these qualities are now measured by formant frequency).
While there is some overlap of individual vowels, you can see that the definitions indicate distinct sets of vowels and that cardinal vowels describe a specific function that general definitions of vowels don't carry: their function is to describe any language by three uniform qualities (height, backness, roundedness).
The eight primary cardinal vowels occupy the corners and four equidistant points with four cardinals on either vertical leg of the quadrilateral. The eight secondary cardinals form pairs with each of the primaries (for this discussion, ignore the diagrammed vowels that are not paired with a primary). While there is some overlap of cardinals with pure vowels, the definitions maintain the cardinal vowels' distinction of function.
The seven checked steady-state pure vowels are represented by the standard phonetic keywords:
The six secondary, or free steady-state, pure vowels are represented by the standard phonetic keywords: