Derived versions of differential topology are becoming prominent tools in symplectic geometry. Whether or not you think of them via topoi is not crucial (I certainly can't), and perhaps the terminology turns off more people than it draws, but these ideas are being put to serious use by very serious no-nonsense mathematicians -- I think an excellent (though of course not isolated) example is the work of differential geometer Dominic Joyce who explains beautifully the necessities that led him deep into this area, see his 800 page book project on D-manifolds (which admittedly adapts a truncated version of the $\infty$-world for concreteness but is undoubtedly part of this story.
One way to express (very briefly) the issues is to say the derived (or $\infty$) language allows one to bypass the geometric but very subtle issues of transversality which seriously interfere with progress in some areas of geometry (Floer theory). Intersections, fiber products, and other constructions arising in moduli theory (obstructions/virtual fundamental classes) naturally lead
to derived manifolds, which retain enough structure to allow algebraic constructions to work without the need for establishing and keeping track of perturbations. (This is not my area, so I can't seriously defend the need for this against a skeptic, but Joyce can..) Let me also say that this kind of geometry makes lots of geometric results (like the Atiyah-Bott fixed point theorem, some Grothendieck-Riemann-Roch and index theorems etc) completely formal. That for me is the main draw of this higher language -- it makes math that has a chance to be formal indeed formal. That's not the case for many results (and probably everything I'm saying applies more to differential topology than geometry) but that's when there are large areas where you might have dreamed that elegant abstract constructions might work but reality has proved disappointingly different, it's exciting to see that there are new languages that may (or may not) turn out up to the task.
Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of smooth manifolds say, based on which one defines differentials, and all sorts of de Rham cohomology etc.; I believe the "differential cohomology" is of this sort. On the other hand, there is the notion of the "tangent $\infty$-topos". I vaguely understand the reason for also calling this "tangent", but it is by a series of analogies, and there seems to be no relation between these two concepts in the case at hand.
More specifically, for condensed anima, the tangent $\infty$-topos is simply the $\infty$-category of pairs $(X,A)$ where $X$ is a condensed anima and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a hypercomplete sheaf of spectra on the site of condensed sets over $X$. This is definitely a very interesting structure. This whole concept of $6$-functor formalisms is very much about such categories. Working with torsion coefficients, and allowing only "relatively discrete" coefficients, I've developed something along those lines in Etale cohomology of diamonds. Something even closer is in Chapter VII of Geometrization of the local Langlands correspondence (link should be active in a few days is active), where we restrict to the solid objects in $\mathcal D(\mathrm{Cond}_{/X},\mathbb Z_\ell)$. A critical role is then played by the left adjoint $f_\natural$ to pullback $f^\ast$. These do not exist in any classical setup, but have excellent formal properties. In fact, one gets a variant of a $6$-functor formalism where homology and cohomology are now on equal footing again (and arguably homology is even more primitive, again): The pullback functor $f^\ast$ admits a left adjoint $f_\natural$ ("homology") and a right adjoint $Rf_\ast$ ("cohomology"), both of which commute with any pullback. Moreover, Poincare duality holds for proper smooth maps. So yes, there's something interesting about this.
On the other hand, the reference to the differential cohomology hexagon in the question confuses me. For condensed sets, there are no tangent spaces, no differential forms, etc., and you can't get them back by magic. I think the problem is that I have no idea what the term "differential cohesion" means, but my strong feeling is that to have "differential cohesion" one needs extra structure like tangent spaces on the model spaces (and that passing to the "tangent $\infty$-topos" is not at all supplying these, as this is a very different procedure).
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Edit: Thanks to the David's for their enlightening comments! Now I understand that the part with the differentials is really only in the examples, not in the "differential cohomology hexagon". For my convenience, let me reformulate this hexagon in my own language.
Say $X$ is a condensed anima, and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a sheaf of spectra on condensed sets over $X$. (Or take $X$ a scheme, and $A\in \mathcal D(X_{\mathrm{proét}},\mathbb Z_\ell)$, or $X$ a small v-stack and $A\in \mathcal D(X_v,\mathbb Z_\ell)$, or...) Let $\pi$ denote the projection from the site of $X$ to the (pro-étale) site of the point. Then pullback $\pi^\ast$ has a right adjoint $R\pi_\ast$ ("cohomology") and a left adjoint that I will denote $\pi_\natural$ ("homology").
In particular, we get condensed spectra $R\pi_\ast A$ (=$R\Gamma(X,A)$), the cohomology of $A$, and $\pi_\natural A$, the homology of $A$. By adjunction, we get maps
$$
\pi^\ast R\pi_\ast A\to A\to \pi^\ast \pi_\natural A.
$$
Let $\overline{A}=\mathrm{cofib}(\pi^\ast R\pi_\ast A\to A)$ and $\tilde{A}=\mathrm{fib}(A\to \pi^\ast \pi_\natural A)$. Then there is a pullback square
$$\begin{CD}
A @>>> \overline{A}\\@VVV @VVV\\ \pi^\ast \pi_\natural A @>>> \pi^\ast \pi_\natural \overline{A}
\end{CD}
$$
and a pushout square
$$\begin{CD}
\pi^\ast R\pi_\ast \tilde{A} @>>> \pi^\ast R\pi_\ast A\\@VVV @VVV\\ \tilde{A} @>>> A
\end{CD}
$$
Of course, pushout squares and pullback squares are equivalent, but I want to stress that one wants to use them to recover $A$ from "simpler" information. However, to me $\tilde{A}\to A\to \overline{A}$ all feel extremely similar, and I'd regard these squares as simple statements about how to analyze the small difference between them.
Best Answer
If we consider just shape modality "$\Pi$" (or "ʃ") and flat modality $\flat$ (which are sufficient for the differential cohomology hexagon, then the traditional arithmetic fracture squares are the left half of the differential hexagon for a kind of pointed $E_\infty$-derived cohesive geometry modeled on Greenlees-May duality. For more on this see my talk Differential cohesion and Idelic structure.
If we consider in addition also the sharp modality $\sharp$ (which is needed to construct moduli stacks of differential cocycles) then an "exotic" model for cohesive homotopy theory is global equivariant homotopy theory. This was found by Charkes Rezk, see his writeup "Global Homotopy Theory and Cohesion".
For these "exotic" models shape is not the localization at an interval object, and hence these exotic differential cohomology theories need not satisy the fundamental theorem of calculus/Stokes theorem (nLab:Stokes theorem -- Abstract formulation in cohesion)
By the way, even if the "standard model" with interval object the real line feels very standard, there are things still to be learned here. Readers with an interest in formal logic might enjoy Mike Shulman's recent "Brouwer's fixed-point theorem in real-cohesive homotopy type theory".
A mild but important variant of the standard model over smooth manifolds which still does satisfy the integration theorem is the model over complex analytic manifolds (see at complex analytic infinity-groupoid). In here ordinary differential cohomology comes out as the theory of holomorphic $p$-gerbes with connection (which of course was what Deligne considered first, back in 1971.) This is also the context of Hopkins-Quick 12.
Then, both the model over smooth and over complex-analytic manifolds may be refined to supergeometry. When doing so, then the adjoint triple of cohesion is accompanied by a progression of two more adjoint triples, each "resolving" the former one. The second one gives what I had called "differential cohesion", which serves to axiomatized manifolds and PDE theory. The third reflects the two possible ways of projecting out bosonic formal geometry from supergeometry. This gives a second cohesive structure on supergeometric homotopy theory.
Details for this super-cohesion are in v2 of my book, some key points are in nLab:super Cartan geometry; exposition includes Modern physics formalized in Modal homotopy theory. Maybe see also the chapter Higher prequantum geometry which I am preparing for "New Spaces in Mathematics and Physics".