We know (for now about 40 years, thank Adleman, Bennet and Gill) that the inclusion BPP $\subseteq$ P/poly, and an even stronger BPP/poly $\subseteq$ P/poly hold. The "/poly" means that we work non-uniformly (a separate circuit for each input length $n$), while P without this "/poly" means we have one Turing machine for all possible input lengths $n$, even longer than, say, $n$ = the number of seconds to the next "Big Bang".
Question 1: What new would a proof (or disproof) of BPP = P contribute to our knowledge after we know BPP $\subseteq$ P/poly?
Under "new" I mean any really surprising consequences, like collapse/separation of other complexity classes. Compare this with consequences the proof/disproof of NP $\subseteq$ P/poly would deliver.
[ADDED 08.10.2017]: One really surprising consequence of BPP $\not\subseteq$ P would be that, as shown by Impagliazzo and Wigderson, all(!) problems in E = DTIME$[2^{O(n)}]$ would have circuits of size $2^{o(n)}$. Thanks to Ryan for recalling this result.
Question 2: Why we cannot prove BPP = P along similar lines as the proof of BPP/poly $\subseteq$ P/poly?
One "obvious" obstacle is finite vs. infinite domain issue: boolean circuits work over finite domains, whereas Turing machines work over entire set $\{0,1\}^*$ of $0$-$1$ strings of any length. So, to derandomize probabilistic boolean circuits, it is enough to take the majority of independent copies of a probabilistic circuit, and to apply Chernoff's inequality, together with the union bound. Of course, over infinite domains, this simple majority rule won't work.
But is this (infinite domain) a real "obstacle"? By using results from statistical learning theory (VC dimension), we already can prove that BPP/poly $\subseteq$ P/poly holds also for circuits working over infinite domains, like arithmetic circuits (working over all real numbers); see e.g. this paper of Cucker at al. When using a similar approach, all we would need is to show that the VC dimension of poly-time Turing machines cannot be too large. Has anybody seen any attempts to make this latter step?
NOTE [added 07.10.2017]: In the context of derandomization, the VC dimension of a class $F$ of functions $f:X\to Y$ is defined as the maximum number $v$ for which there are functions $f_1,\ldots,f_v$ in $F$ such that for every $S\subseteq\{1,\ldots,v\}$ there is a point $(x,y)\in X\times Y$ with $f_i(x)=y$ iff $i\in S$. I.e. we shatter not the sets of points via functions but rather sets of functions via points. (The two resulting definitions of the VC dimension are related, but exponentially.)
The results (known as uniform convergence in probability) then imply the following: if for each input $x\in X$, a randomly picked function $\mathbf{f}\in F$ (under some probability distribution on $F$) satisfies $\mathrm{Prob}\{\mathbf{f}(x)=f(x)\}\geq 1/2+c$ for a constant $c>0$, then $f(x)$ can be computed on all inputs $x\in X$ as a majority of some $m=O(v)$ (fixed) functions from $F$. See, e.g. Corollary 2 in Haussler's paper. [For this to hold, there are some mild measurability conditions on $F$.]
For example, if $F$ is the set of all polynomials $f:\mathbb{R}^n\to\mathbb{R}$ computable by arithmetic circuits of size $\leq s$, then all polynomials in $F$ have degree at most $D=2^s$. By using known upper bounds on the number of zero-patterns of polynomials (see, e.g. this paper), one can show that the VC dimension of $F$ is $O(n\log D)=O(ns)$. This implies the inclusion BPP/poly $\subseteq$ P/poly for arithmetic circuits.
