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Why do linguists represent the vowels as a vowel diagram, but use a table instead to represent the consonants?

For example, compare a vocal diagram:

against a consonant table:

consonant table

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    Because the major variables in vowels are tongue positions - high/low and front/back, and two-dimensional map of the mouth indicates those well enough. But there are many more major variables for consonants, and most of them are not relevant to all consonants, so a two-dimensional map isn't possible. Therefore a table with labelled columns and rows (and a lot of empty spaces, because of that lack of general relevance) is more useful.
    – jlawler
    Mar 7 '17 at 19:38
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    @jlawler, your comment constitutes for an answer.
    – bytebuster
    Mar 7 '17 at 19:44
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The vowel diagram shows the position of the tongue when pronouncing this or that vowel. This position, as well as lips being rounded/unrounded are the only factors that influence the quality of the English vowels (the diagram is for the English language), all the vowels are produced in the same manner: the air freely flows through the open mouth, the position of the tongue only changes the shape and the volume of the resonator (mouth cavity).

But with the consonants it is all much more complicated. The quality of the consonants is not defined merely by the tongue position as is the case with the vowels. Some consonants are produced without any use of the tongue whatsoever (bilabial, labiodental, glottal), there are several manners in which the consonants are produced (stops, fricatives, affricates) as opposed to just one manner with the vowels. Also, while pronouncing a consonant, the air can flow not only through the mouth, as with the vowels, but also through the nose (nasal consonants). And on top of that, the consonants can be voiced and voiceless. All of this makes the system of the consonants much more multi-dimensional than the two-dimentional vowel system. In other words, a vowel is defined by just 2 factors, tongue position and rounded/unrounded lips, a consonant is defined by much more factors that simply cannot be squeezed into a two-dimentional diagram.

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To add to the 2 answers, it seems worth noting that the "highness" and "backness" of the two dimensional vowel space can be mapped to the frequencies of formants one and two of a vowel articulation.

The general answer is that consonants have too many distinctive features to give a nice spatial representation like what we have with vowels.

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The differences in presentation reflect the nature of how consonants vs. vowels are produced. The arrangement on the horizontal axis reflects location of the major constriction, but is mapped in terms of the entire vocal tract as a straightened tube for consonants, vs. just the tongue body for vowels. The vertical axis for vowels iconically represents degree of constriction, but for consonants that is only half-true (plosives and fricatives). Since there are separate symbols for nasal or phonatory differences in consonants (and not in vowels), the vertical dimension has to be recruited to also represent those consonant properties. There are more degrees of constriction for vowels compared to consonants.

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