It influences your next game negatively if you lose.
I think the rival game matters only after it has been played- It gives you a boost if you win and a drop if you lose. I think it is the same as the TV games or the cup games. They all affect your fan survey, which is some sort of modifier to the attendance calculator. I seem to remember a note about how the different factors are not weighted equally, though I can't be certain. If they are not weighted equally, I don't know which ones are more important than others, but I would like to...
Once I scored a basket that still makes me laugh.