Redox potential of a lead–acid battery
The potentials depend on the form of the compounds. 
It is true that in solution 

$\ce{Pb^{2+}  (aq) + 2e^- -&gt; Pb (s)} $ 

is -0.126 V. 

But in the case of a battery we have:

$\ce{PbSO4 (s) + 2e^- -&gt; Pb (s) + SO4^{2-} (aq)}$

And in this case the $\ce{Pb^{2+}}$ is in solid form and the potential is -0.356 V. 

In a battery the sulphate is insoluble and it is required that it sticks to the electrode, otherwise the reverse reaction can not occur. 

A table of potentials can be found here