There are many different crampon geometries. While the exact choice comes down to personal preference, let's discuss the canonical choices and intended uses. I will limit my discussion to automatic/clamping crampons intended for climbing, ignoring discussion of strap-on crampons for general mountaineering and bolt-on crampons for high-end mixed climbing.
Horizontal vs Vertical Frontpoints
The first design bifurcation is that of horizontal front-points (e.g., Petzl Irvis, BD Sabretooth, Grivel Haute Route) versus vertical front-points. Horizontal front-points significantly outperform vertical front-points on steep snow and neve, with the points acting almost like miniature snowshoes, compared to vertical front points which tend to slice through the snow.
Horizontal front-points are generally the best choice for technical mountaineering and alpinism. Vertical front-points generally perform much better on pure ice due to deeper penetration, rigidity, and holding power once placed. This is not exclusionary---horizontal front-points can climb ice (especially lower angle alpine ice) quite well and vertical front-points can climb on snow and neve (albeit at the expense of more effort kicking and less security).
Some manufacturers attempt the best of both worlds by adding a horizontal gusset to the back of a vertical front-point (e.g., Black Diamond refers to this as a "hooded front-point"), with the idea of having vertical front points for ice and the gusset engaging snow when penetrated deeply.
Horizontal front-points can also have a niche use case for exceedingly thin smears of ice over rock, as they can almost chop out a miniature ledge in ice that is too thin to get any penetration with vertical front-points.
Dual vs Mono Front-Points
The distinction between dual and mono front-point setups is more subtle than the difference between horizontal and vertical front-points. Further muddying the distinction are crampons that have a dominant front-point and a smaller secondary front-point (e.g., Grivel Rambo) and adjustable setups (e.g., the crampons pictured in the OP). We'll restrict our discussion to mono vs dual setups.
The basic choice between mono and dual front-points is generally that of precision (mono) versus stability (dual), with the exact ice conditions (and personal preference) dictating which attribute is more desirable. Many of the hardest ice climbs in the world have been done with both monos and duals.
The precision of mono front-points excel on irregular terrain, the paradigm setting example being that of mixed climbing / dry tooling. By having only a single point, a mono crampon allows for precisely slotting into pockets, cracks, fissures, and small edges on rock. In contrast, the secondary point on dual crampons might entirely prevent the crampon from securely using these features. This distinction also carries over to highly featured, chandeliered and candle-sticked ice, where dual front-points might require seeking out small patches of uniform ice. Mono front-points can also be useful on steep uniform ice by allowing you to "draft" your former ice tool placements with your mono front-point. This can be especially beneficial for very dense ice (where kicking out a placement involves significant effort).
I also find mono points beneficial in very brittle ice that is prone to "dinner plating," as the single point tends to disturb less ice and allow fractured ice to easily exit to the side. In contrast, dual front-points tend to break more of this brittle ice---especially the ice between the front points. The counter-argument is that dual front-points allow for a more secure placement in these tenuously brittle conditions once a suitable placement has been excavated.
The conditions that dual front-points excel in would be something like a uniform flow of ice in plastic hero conditions. The amount of stability gained by having two front points (especially in resistance to twisting) can be quite significant, resulting in better security and less fatigue. Dual points also spread the load over more of the ice, which can be quite useful for thin smears and steep snow / neve (although not quite as well as horizontal front-points).
Dual front-points might be preferred for a day of WI3--4 in good conditions or a long couloir climb that involves steep neve and long sections of AI 2--3. I also think that the added stability of dual points is much easier to learn with and much more confidence inspiring.
Mono front-points might be preferred for a route that involves any extensive mixed climbing (perhaps strongly preferred) or hard routes that have formed with extensive candle-sticking.
