The Contagonist represents specific aspects of the human mind (Temptation, Hinder), so the character that represents your Contagonist within the story will be the one where we see these elements explored. In fact I think the archetype players are described as being a function of these elements such that the Contagonist should always be tempting or hindering others (as opposed to simply being tempted and hindered himself).
The Influence Character is a perspective of the Storymind. As such, it does not represent any specific element of the mind (such as Temptation or Hinder). Instead, it represents the You perspective. This means that the Storymind will look at the Influnce Character perspective as representing a singular effort to solve the problem outside of the Main Character. The Influence Character is seen from the perspective of the Main Character (or the personal I perspective). This means the Storymind will look to the MC to ask 'How do I solve this problem?' and it looks to the IC to ask, 'How are you solving this problem?' The influence that the Influence Character has, then, seems to come from the MC perspective deciding if it should continue solving the problem the way it always has (Steadfast) or by adopting the IC's method (Change).
It could be that the IC is also the Contagonist player, but this is not necessary. If they are found in the same character, the way they would work together would be that the MC looks at how the IC solves the problem and finds that the character, at least within the OS Throughline, Tempts and Hinders others.
Where I get a little fuzzy is in trying to keep the ICs influence within the IC throughline while making sure they represent the correct elements or Archetype in the OS. Because that is also a part of how they 'work together'. The Contagonist (and the seven other Archetypes) are generally spoken of strictly as they relate to the OS. The Influence Character perspective is strictly about the IC story.