Consequences of #P = FP
Here are a few theoretical consequences of the equality FP=#P, although they have nothing to do with artificial intelligence.  The assumption FP=#P is equivalent to P=PP, so let me use the latter notation.

If P=PP, then we have P=BQP: quantum polynomial-time computation can be simulated by classical, deterministic polynomial-time computation.  This is a direct consequence of BQP⊆PP [ADH97, FR98] (and of an earlier result BQP⊆PPP [BV97]).  On top of my knowledge, P=BQP is not known to follow from the assumption P=NP.  This situation is different from the case of randomized computation (BPP): since BPP⊆NPNP [Lau83], the equality P=BPP follows from P=NP.

Another consequence of P=PP is that the Blum-Shub-Smale model of computation over the reals with rational constants is equvalent to Turing machines in a certain sense.  More precisely, P=PP implies P=BP(Pℝ0); that is, if a language L⊆{0,1}* is decidable by a constant-free program over the reals in polynomial time, then L is decidable by a polynomial-time Turing machine.  (Here “BP” stands for “Boolean part” and has nothing to do with BPP.)  This follows from BP(Pℝ0)⊆CH [ABKM09].  See the paper for definitions.  An important problem in BP(Pℝ0) is the square-root sum problem and friends (e.g. “Given an integer k and a finite set of integer-coordinate points on the plane, is there a spanning tree of total length at most k?”) [Tiw92].

Similarly to the second argument, the problem of computing a specific bit in xy when positive integers x and y are given in binary will be in P if P=PP.

References

[ABKM09] Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen and Peter Bro Miltersen.  On the complexity of numerical analysis.  SIAM Journal on Computing, 38(5):1987–2006, Jan. 2009.  http://dx.doi.org/10.1137/070697926

[ADH97] Leonard M. Adleman, Jonathan DeMarrais and Ming-Deh A. Huang.  Quantum computability.  SIAM Journal on Computing, 26(5):1524–1540, Oct. 1997.  http://dx.doi.org/10.1137/S0097539795293639

[BV97] Ethan Bernstein and Umesh Vazirani.  Quantum complexity theory.  SIAM Journal on Computing, 26(5):1411–1473, Oct. 1997.  http://dx.doi.org/10.1137/S0097539796300921

[FR98] Lance Fortnow and John Rogers.  Complexity limitations on quantum computation.  Journal of Computer and System Sciences, 59(2):240–252, Oct. 1999.  http://dx.doi.org/10.1006/jcss.1999.1651

[Lau83] Clemens Lautemann.  BPP and the polynomial time hierarchy.  Information Processing Letters, 17(4):215–217, Nov. 1983.  http://dx.doi.org/10.1016/0020-0190(83)90044-3

[Tiw92] Prasoon Tiwari.  A problem that is easier to solve on the unit-cost algebraic RAM.  Journal of Complexity, 8(4):393–397, Dec. 1992.  http://dx.doi.org/10.1016/0885-064X(92)90003-T