The higher the Senate's favour, the more likely you are to have family members voted into one of the various Senate offices, which all give nice stat benefits, plus in some cases can enable you to avoid penalties for failed missions.

The improvement to your leader's bodyguard that you gain from the highest level of governing building your faction can build is the number of men in the bodyguard. The upper limit to the bodyguard otherwise seems to be determined by the leader's stats and retinue.