Fares can certainly differ significantly on both website. As both airlines have different inventory, even on the same flights.
For example: CDG - AMS - NWI on 13th of August with 18:35 departure on KLM prices at 305 euros, while AirFrance prices the exact same flights at 200 euros. I could list many others.
On award flights, what I see differ the most are the tax amounts.