These two diagrams form what mathematicians might call a dual graph, in that they are sort of mirror images of each other. I'll describe this briefly for the benefit of others.
In the first image, the nodes represent pitches (well, pitch classes), while in the second image, the nodes represent chords. To construct the second image from the first, place a point in the center of each triangle, label it with the name of the chord formed by the surrounding three pitches, and connect it to the centers of the three neighboring triangles. To construct the first image from the second, simply repeat this process: place a point in the center of each hexagon, label it with the name of the pitch that is common to the six surrounding chords, and connect it to the centers of the six neighboring hexagons. For example, the hexagon (c-C-a-F-f-A♭) represents the pitch C, and if you go one hexagon above it (g-G-e-C-c-E♭) this represents the pitch G. The line connecting these two hexagons is the same as the edge C-G in the first image.
Thus all of the intervalic relationships in the first diagram exist in the second diagram, but in a slightly less easy-to-see form (lines between centers of shapes rather than edges). Similarly, in the second diagram, the edges represent transitions between two chords that share two notes in common. But this same information can be seen in the first diagram by crossing an edge from one triangle to an adjacent one.
You might think that, since the second diagram focuses more on chords (by making them be the nodes), it would be better-suited for analyzing chord progressions. However, here we see the downside of the second diagram: not all chords are represented in the diagram, only major and minor triads. There are no diminished or augmented chords, much less suspensions, or sevenths, or other combinations. Furthermore, if you look at any pair of adjacent nodes as being a chord progression, they are always chords that differ by a single note.
On the other hand the first diagram is capable of displaying any possible combination of tones, regardless of whether it is a major or minor triad. However, chord progressions do become rather abstract, as one shape morphs into another.
- Nodes: All pitch classes
- Edges: Intervals, but only consonant ones (m3/M6, M3/m6, P4/P5)
- Shapes: Chords (any possible combination of nodes)
- Nodes: Chords, but only major/minor triads.
- Edges: Chord progressions, but only if they share two common tones.
- Shapes: Pitch classes (common tone of surrounding chords).
Given the clear limitations of the hexagon based diagram (which are easily overcome in the triangle diagram) I'm not sure why someone would choose it over the triangle diagram, given the choice (historically, I think it was developed first, though). Honestly, I'm having a hard time thinking of any cases where you can clearly and easily do something on the hex grid that you absolutely can't do on the triangle.
This agrees with the fact that it's more natural to think of a chord as a combination of multiple pitches, rather than to think of a pitch as the overlapping of multiple chords.