The OP asks, “In some ways faith and believing may mean the same, but is there a way, or ways here, in which pistis and pisteuon are not the same?”
The noun pistis (πίστις G4102), meaning faith, belief, trust, confidence, fidelity is the root of the verb pisteuó (πιστεύω G4100), meaning to believe, to have faith, to trust. It’s not that the words have different meanings, but that they are different parts of speech. One reason why the author would use both a noun and its cognate verb in the same context is to distinguish between the thing (faith) from the action with which it is associated (believing).
In context, the author associates “our faith” less with our act of believing than God’s act of begetting.
1 John 5:1 ESV (parenthesis added)
Everyone who believes 4100(present participle active) that
Jesus is the Christ has been born (perfect indicative middle/passive)
of God.
The participle translated in the ESV as “believes” is actually a verbal adjective. Its use places emphasis on the characteristic rather than the action of those who believe. Furthermore, “believes” is in the present tense while “has been born” is in the perfect tense, indicating past action with ongoing effect. It is God’s action that brings about the victory over the world. In other words, it is God’s action that brings about our faith (cf. Eph 2:8-10).
1 John 5:4
For everyone who has been born (perfect indicative middle/passive) of
God overcomes (present indicative active) the world. And this is the
victory that has overcome the world—our faith4102.