The difficulty level for the battle map determines bonuses given to the troops. On easy, I think your troops get a +4 to attack/defense stats. On medium, there are no bonuses. On hard, it's +4 to the enemy's troops, and very hard is +7 to the enemy's troops. I can't remember which stats get adjusted, maybe morale, attack, and defense, there are a couple of threads that talk about this. In M:TW, the difficulty level also opened up the AI's combat strategy (it would try certain flanking moves, hiding and ambushes, feigned retreats, etc.) but I'm not sure R:TW does this.

The difficulty level on the strategic map affects money matters, maybe diplomacy as well. All auto-resolved battles (including naval combat) are done using the strategic map difficulty, not the battle map difficulty, so your ships tend to die easier.

I think Frogbeastegg's guide has some of the specifics:
https://forums.totalwar.org/vb/showthread.php?t=38382