Bits required to encode difference between number of subgraphs with odd number of edges and number of subgraphs with even number of edges
It is the case that $b \notin o( m )$.

Let $G = ( V_G, E_G )$ be a $3$-regular connected graph. Without loss of generality, let us fix $w = |V_G|$ and focus on $\Delta_{|V_G|} = O_{|V_G|} - E_{|V_G|}$.

Pick an arbitrary edge $e = \{u, v\} \in E_G$. Remove $e$ and replace it with the following gadget:



This has the effect of multiplying $\Delta_{|V_G|}$ by $8$. The resulting graph $H = (V_H, E_H)$ is still $3$-regular connected, and has $60$ vertices more than $G$. 

By repeating this process $i$ times, we obtain a $3$-regular connected graph having $|V_H| = |V_G| + 60i$ vertices, and whose $\Delta_{|V_H|}$ is $8^i$ times bigger than $\Delta_{|V_G|}$. 

After $i = \frac{|V_G|}{3}$ iterations, our final graph will have $|V_H| = 21|V_G|$ vertices and a $\Delta_{|V_H|}$ which is $2^{|V_G|}$ times bigger than the $\Delta_{|V_G|}$ we started with. Let $n = |V_H|$, recall how $m = 1.5n$ for $3$-regular graphs, and we get a $\Delta_{|V_H|}$ whose absolute value is lower-bounded by $2^{\frac{m}{31.5}}$. Henceforth $b$ is linear in $m$ in the worst case, and thus it cannot belong to $o( m )$.

Contrarily to intuition (at least mine), the number of edge covers of odd cardinality and the number of edge covers of even cardinality may diverge by an exponential amount.