Why does the parent connection matter during a NEAT crossover when both connections are identical? I've been reading through NEAT (NeuroEvolution of Augmenting Topologies)'s whitepaper, and one thing is particularly confusing:

Innovation numbers are supposedly used to track the origin of a mutation, so that crossing over is easily done. From what I understand, two identical connections should always have the same innovation number, because their numbers are indexed. However, according to this illustration:



when the two genomes being crossed over share connections with the same innovation number, the resulting innovation is taken from the more fit parent (or random if they are equal fitness).

If the two connections are identical (they have the same innovation number) then why does it matter which parent it comes from, since the connection is the same either way? Or is there something about them that is not the same and I am understanding this wrong?
The genes shown in the diagrams have been simplified so that they only show topology and innovation index number. In addition, the full genome contains weights for each connection and bias for each node. 

The innovation numbers are not tracking changes with connection weights or biases, just the rough time of inserting a specific connection between two nodes. So each parent can have matching connections/innovations, but with different weights associated. The full genome is still subject to cross-over and mutation, the excerpt you quote helps explain how the genome is aligned between parents with different architectures, so that GA crossover and mutation can occur in each connection and node.

Which parent is used to create each offspring gene matters because of the additional connection weight data associated with each parent's gene.