Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become relevant, given that these do not appear in the physics which motivates mirror symmetry?) I would especially appreciate nontechnical answers.
[Math] mirror symmetry with algebraic geometry
ag.algebraic-geometrymirror-symmetrymp.mathematical-physics
Related Solutions
For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $\beta:\operatorname{Pic}(X)\to\mathbb Z$, the space $\overline{M}_{g,n}(X,\beta)$ of stable maps is well defined as an Artin stack with finite stabilizer, no matter the characteristic of $k$. You can even replace $k$ by $\mathbb Z$ if you like.
Now if $X$ is a smooth projective scheme over $R=\mathbb Z[1/N]$ for some integer $N$, then $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ is a Deligne-Mumford stack for almost all primes $p$. For such $p$, $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ has a virtual fundamental cycle, and so you have well-defined Gromov-Witten invariants. This holds for all but finitely many $p$. Nothing about $\mathbb C$ here, that is my point, the construction is purely algebraic and very general.
It is when you say "Hodge structures" then you better work over $\mathbb C$, unless you mean $p$-Hodge structures.
As far as mirror symmetry in characteristic $p$, much of it is again characteristic-free. For example Batyrev's combinatorial mirror symmetry for Calabi-Yau hypersurfaces in toric varieties is simply the duality between reflexive polytopes. You can do that in any characteristic, indeed over $\mathbb Z$ if you like.
Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab. In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)
Best Answer
Here are a few scattered observations:
Our ability to construct examples (e.g. of CY manifolds) is limited, and the tools of algebraic geometry are perfectly suited to doing so (as has been noted).
Toric varieties are a source of many examples -- Batyrev-Borisov pairs -- and they are even "more" than algebraic, they're combinatorial. In fact, the whole business is really about integers in the end, so combinatorics reigns supreme.
The fuzziness of $A_\infty$ structures is more suited to algebraic topology rather than geometry.
Continuity of certain structures (which are created from counting problems) across walls, scores some points for analysis over combinatorics and algebra.
Elliptic curves are only "kinda" algebraic, and the mirror phenomenon there is certainly transcendental.
Physics indeed does not care too much about how the spaces are constructed, but (as has been noted) even the non-topological version of mirror symmetry is an equivalence of a very algebraic structure (which includes representations of superconformal algebras).
I was hoping to unify these idle thoughts into a coherent response, but I don't think I can. Maybe the algebraic geometric aspects just grew faster because the mathematics is "easier" (or at least better understood by more mathematicians): witness the slow uptake of BCOV and its antiholomorphicity within mathematics.
To respond personally: these days, I try to transfer the algebraic and symplectic structures to combinatorics so that I can hold them in my hand and try to understand them better.