Aren't two infinite asymmetric graphs always identical?

Question

Suppose you have an infinite graph $G$. I assume $G$ to be cubic and planar. No further conditions, so it will be asymmetric, maybe in the sense of cubic planar version of Rado's graph: Every possible combination of faces exists somewhere in this graph (like in normal numbers).

Now you are given a second one of such a graph $G'$. They might look different at first glance, but aren't they actually identical?

Answer 1

These are not isomorphic: the second has bridges, and the first has none.

          --o----o----o----o--  
            |    |    |    |  
          --o----o----o----o--

            o          o          o  
           /|\        /|\        /|\  
          / | \      / | \      / | \  
       --o  |  o----o  |  o----o  |  o--  
          \ | /      \ | /      \ | /  
           \|/        \|/        \|/  
            0          o          o